Time-Decayed Vector Search in the Rhythm of TANGO: Jointly Modeling Semantic Similarity and Temporal Freshness

arXiv:2609.00548v1 [cs.DB] 1 Sep 2026

Time-Decayed Vector Search in the Rhythm of TANGO: Jointly Modeling Semantic Similarity and Temporal Freshness Jiuqi Wei

Qiyao Luo

Quanqing Xu

OceanBase, Ant Group weijiuqi.wjq@antgroup.com

OceanBase, Ant Group luoqiyao.lqy@antgroup.com

OceanBase, Ant Group xuquanqing.xqq@oceanbase.com

Chuanhui Yang

Themis Palpanas

OceanBase, Ant Group rizhao.ych@oceanbase.com

LIPADE, Université Paris Cité themis@mi.parisdescartes.fr

ABSTRACT

1

Vector search typically measures relevance through semantic similarity under a fixed scoring function. However, in a growing range of applications, relevance may evolve over time, making temporal freshness an additional signal beyond semantic similarity. In this paper, we formalize time-decayed vector search (TDVS), which incorporates continuous temporal decay into the search objective so that relevance is jointly determined by semantic similarity and temporal freshness. We design Score-Preserving Temporal Reduction (STR) that enables existing Maximum Inner Product Search indexes to directly support TDVS. We further present Chronos, a TDVSnative framework that derives an exact metric formulation and introduces Query-Orthogonal TimeLift to control data–data geometry while preserving all query–data scores and rankings. Building on Chronos, we propose TANGO, a hierarchical graph index that adopts layer-specific TimeLift geometries to preserve temporal locality at the base layer while strengthening long-range semantic connectivity in upper layers. TANGO traverses the hierarchy using the exact TDVS score, caches temporal factors to reduce computation, and supports efficient online insertion. Extensive experiments show that TANGO achieves up to 3.5× higher query throughput and 4.05× faster index construction than state-of-the-art graphbased competitors. TANGO also maintains its advantage over all competitors across diverse temporal settings and enables efficient online insertion, demonstrating its robustness and practicality.

Vector search has become a fundamental building block of modern data management and AI systems, supporting a wide range of applications such as semantic search, retrieval-augmented generation (RAG), large language models (LLMs), recommender systems, and agentic workflows [10, 55]. In its standard form, vector search returns the items that are most similar to a query under a fixed similarity function, such as inner product, cosine similarity, or Euclidean distance [2, 50]. This formulation works well in many cases, but it assumes that relevance depends only on semantic similarity and remains unchanged once an item’s embedding is fixed. Such an assumption no longer holds in many emerging applications, where temporal freshness is becoming an increasingly important factor in determining relevance [33, 47, 56]. Emerging Scenarios. Consider several representative examples. In agent memory [32, 34], an agent may store a long history of past interactions, observations, and intermediate reasoning results. For a new query, an old memory may be semantically very similar, yet still be less useful than a more recent one that better reflects the agent’s current state, goals, or environment. In freshness-sensitive RAG [39] and temporal question answering [21], older passages may still retain partial utility, but should gradually become less competitive than more recent passages as the context evolves. In recencysensitive web search [9], the relevance of a page may change on the scale of days or even hours for rapidly evolving events, requiring search results to account for both topical relevance and freshness. In dynamic content recommendation [42], older items may still match user interests, but should often be ranked below newer items whose utility is more aligned with current attention and demand. This demand is also reflected in the industry: Qdrant, for example, supports exponential time-based score boosting for news and other freshness-sensitive searches [38]. Existing Time-Aware Pipelines. Existing time-aware vectorsearch approaches mainly follow two paradigms. The first is filterbased search, where time is represented as metadata and range filters are applied before or during search [18, 49]. In this design, temporal information is used mainly to restrict the search space, while the underlying vector search objective remains unchanged and continues to rank candidates purely by semantic similarity. The second is rerank-based search, which performs semantic search first and then incorporates temporal signals through reranking or score fusion [9, 32]. Such methods retain a standard semantic retriever as the backbone and introduce time only after an initial candidate set has been produced. Overall, current vector search solutions still

PVLDB Reference Format: Jiuqi Wei, Qiyao Luo, Quanqing Xu, Chuanhui Yang, and Themis Palpanas. Time-Decayed Vector Search in the Rhythm of TANGO: Jointly Modeling Semantic Similarity and Temporal Freshness . PVLDB, 20(1): XXX-XXX, 2027. doi:XX.XX/XXX.XX PVLDB Artifact Availability: The source code, data, and/or other artifacts have been made available at https://github.com/WeiJiuQi/Chronos.

This work is licensed under the Creative Commons BY-NC-ND 4.0 International License. Visit https://creativecommons.org/licenses/by-nc-nd/4.0/ to view a copy of this license. For any use beyond those covered by this license, obtain permission by emailing info@vldb.org. Copyright is held by the owner/author(s). Publication rights licensed to the VLDB Endowment. Proceedings of the VLDB Endowment, Vol. 20, No. 1 ISSN 2150-8097. doi:XX.XX/XXX.XX

INTRODUCTION

Table 1: Summary of key notations.

treat time as an auxiliary factor layered on top of semantic search, rather than as a native component of the search objective. Limitations and Motivation. Despite these efforts, existing vector search solutions remain inadequate for dynamic workloads in which relevance derived from semantic similarity changes continuously over time. Filter-based methods are too coarse to capture soft temporal decay, since relevance often decreases gradually rather than disappearing at a fixed timestamp threshold. Rerank-based pipelines are limited by the recall of the first-stage retriever: if an item with the right semantic–temporal trade-off is not retrieved initially, no downstream reranker can recover it. This gap calls for a new formulation and a corresponding system solution. Rather than adding temporal signals to an existing vector search pipeline, we ask a more fundamental and challenging question: how should vector search be reformulated when relevance changes continuously over time? Addressing this question requires incorporating temporal decay directly into the search objective, so that indexing and search can be built around relevance jointly determined by semantics and time. This motivates a new problem that we call time-decayed vector search: given a query vector and a collection of timestamped data vectors, retrieve the items that are most relevant under a scoring function that combines semantic similarity with time decay. Our Solution. In this paper, we formalize Time-Decayed Vector Search (TDVS) as a new vector search problem and present Chronos, a TDVS-native framework, together with its hierarchical graph index TANGO to efficiently support TDVS. Specifically, we first formulate TDVS over timestamped vectors by incorporating continuous temporal decay directly into the search objective, so that relevance is jointly determined by semantic similarity and temporal freshness at query time (Section 2.2). We then design two solutions for TDVS: a generic solution Score-Preserving Temporal Reduction (STR) (Section 2.3) and a TDVS-native solution Chronos (Section 3). STR exactly reduces TDVS to conventional Maximum Inner Product Search (MIPS), enabling existing MIPS indexes to be directly reused to support TDVS, but it inherits structural constraints from the temporal factorization. To overcome these constraints, we present the Chronos framework, which derives an exact metric formulation for unit-norm embeddings and introduces Query-Orthogonal TimeLift to control data– data geometry, while preserving all query–data scores and rankings. Building on Chronos, we propose TANGO (Section 4), short for Time-Aware Navigable Graph with Query-Orthogonal TimeLift. TANGO is a TDVS-native hierarchical graph index that assigns different TimeLift geometries to different graph layers, enabling upper layers to strengthen long-range semantic connectivity while the base layer preserves temporal locality. TANGO traverses the hierarchy under the exact TDVS score, caches temporal factors to eliminate repeated exponentiation, and supports efficient online insertion. Extensive experiments on seven real-world datasets demonstrate that TANGO consistently outperforms state-of-theart graph-based competitors in both query and index performance while remaining robust across diverse temporal settings (Section 5). Our contributions are summarized as follows: • We formalize time-decayed vector search (TDVS) and study two representative objectives that combine semantic similarity and

Notation

Description

D, 𝑛, 𝑑 (𝑥𝑖 , 𝑡𝑖 ), (𝑞, 𝜏) 𝑠 (·, ·), Δ𝑖 , 𝑤 (·) 𝜆, ℎ 𝐹𝑚 , 𝛼, 𝑘 𝜓𝑄/𝑋 ,𝑇 , 𝑧𝑖(𝑇 ) , 𝑔𝜏(𝑇 ) 𝑞e𝑚 e𝑖𝑚 𝜏 ,𝑥 𝑘𝜆 , 𝜙 𝜆 , 𝑄𝑚 , 𝑃𝑚 , H𝑚 𝑐 𝑖 𝑗 , 𝑘𝑖 𝑗 𝑑𝑚 , 𝐷𝑚 𝜅, 𝜅 ℓ , 𝑑𝑚(𝜅 ) , 𝐷𝑚(𝜅 ) Gℓ 𝑡 0 , 𝑎𝜏 , 𝑏 𝑖

Dataset, cardinality, and vector dimension Data and query vector–time pairs Semantic similarity, item age, and time-decay function Decay rate and half-life, with 𝜆 = ln 2/ℎ TDVS score, additive semantic weight, and result size STR encoders, anchor, and temporal factors STR mapped query/data vectors Chronos kernel, feature map, mappings, and Hilbert space Pairwise semantic and temporal affinities Hilbert-space metric and its squared form TimeLift parameter, layer value, metric, and squared form TANGO graph at layer ℓ Reference time and cached query/data factors

temporal freshness additively or multiplicatively. We further design Score-Preserving Temporal Reduction (STR), an exact scorepreserving reduction that converts TDVS into Maximum Inner Product Search (MIPS), enabling existing MIPS indexes to directly support TDVS. • We present Chronos, a TDVS-native framework that provides an exact metricization of the complete TDVS objective. Its QueryOrthogonal TimeLift establishes a family of query-equivalent, but geometry-distinct metric spaces, enabling controllable semantic– temporal geometry, while exactly preserving every TDVS score and ranking. • We propose TANGO, a TDVS-native hierarchical graph index built on Chronos. TANGO realizes layer-specific semantic–temporal geometries within a unified graph, jointly supporting temporal locality and long-range semantic connectivity, and at the same time, enabling efficient online insertion. • We conduct extensive experiments comparing TANGO with stateof-the-art graph-based methods. TANGO achieves up to 3.5× higher query throughput and 4.05× faster index construction than the bestperforming competitors. Across diverse temporal settings, TANGO still outperforms all competitors and supports efficient online insertion, demonstrating its robustness and practicality.

2 PROBLEM STATEMENT 2.1 Classical Vector Search We begin with the standard vector search problem. Given a dataset D = {𝑥𝑖 }𝑛𝑖=1 , where 𝑥𝑖 ∈ R𝑑 , a query 𝑞 ∈ R𝑑 , a similarity function 𝑠 (·, ·), and an integer 𝑘, exact vector search returns the 𝑘 data vectors with the highest similarity to the query: N𝑘 (𝑞) = TopK𝑘𝑥𝑖 ∈ D 𝑠 (𝑞, 𝑥𝑖 ). Table 1 summarizes the key notation used throughout the paper. In high-dimensional settings, exact vector search becomes challenging due to the curse of dimensionality [4, 19]; approximate vector search is therefore the more practical choice, returning an apb𝑘 (𝑞) at substantially lower retrieval cost [11, proximate top-𝑘 set N b (𝑞)∩N (𝑞) | |N

12], with answer quality measured by Recall@𝑘 = 𝑘 𝑘 𝑘 . Classical vector search, whether exact or approximate, assumes that relevance is fully determined by semantic similarity and does not change once an item’s embedding is fixed. This assumption is appropriate for many workloads, but it becomes insufficient when the usefulness of an item changes with time [21, 34, 39, 42]. 2

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Figure 1: An illustrative example of time-decayed vector search: (a) candidates’ semantic similarity to query and temporal freshness (with half-life ℎ = 30); (b) and (c) TDVS ranking results: (c) reports the scores and rankings under multiplicative TDVS, and additive (𝛼 = 0.7) TDVS, while (b) visualizes these results in the semantic similarity–freshness space, where yellow, orange, blue, and white denote top-3 membership under both modes, multiplicative only, additive only, and neither, respectively.

2.2

Time-Decayed Vector Search

Definition 4 (Multiplicative Time-Decayed Vector Search). The multiplicative form of TDVS is defined by the scoring function1

We now introduce time-decayed vector search, where each data vector is associated with a timestamp. Formally, let D = {(𝑥𝑖 , 𝑡𝑖 )}𝑛𝑖=1 , where 𝑥𝑖 ∈ R𝑑 is the vector of item 𝑖, and 𝑡𝑖 denotes the time at which the item is created. For a query issued at time 𝜏, we define the age of item 𝑖 as Δ𝑖 (𝜏) = 𝜏 − 𝑡𝑖 , where Δ𝑖 (𝜏) ≥ 0.

𝐹 mul (𝑞, 𝑥𝑖 , 𝜏) = 𝑠 (𝑞, 𝑥𝑖 ) 𝑤 (Δ𝑖 (𝜏)). Note that in practice, the TDVS parameters are straightforward to configure: the half-life ℎ directly specifies the application-specific timescale at which freshness is reduced by half, while the additive weight 𝛼 controls the desired semantic–temporal trade-off. Similar controls are already used in existing systems: OpenClaw adopts a 30-day recency half-life [32], while Qdrant exposes configurable decay timescales and score weights [38]. When relevance feedback is available, ℎ and 𝛼 can also be tuned on held-out queries. Figure 1 illustrates how semantic similarity and temporal freshness jointly affect TDVS ranking (in this example, half-life of ℎ = 30 and additive weight 𝛼 = 0.7). Point 𝐵, which has both high semantic similarity and high freshness, ranks first under both objectives. Multiplicative TDVS promotes the fresher points 𝐹 and 𝐻 into the top-3, while substantially lowering the rank of the highly similar, but old, point 𝐴. In contrast, additive TDVS retains 𝐶 and 𝐴 in the top-3 because semantic similarity receives the larger weight. Figure 1(c) demonstrates that the two TDVS modes are suitable for different freshness requirements. Additive TDVS treats temporal freshness as a complementary preference signal, with 𝛼 controlling the semantic–temporal trade-off. It is therefore appropriate when time matters, but strong semantic matches should remain competitive, such as long-term contextual memories, whose usefulness is not tightly coupled to an agent’s current state [34]. Multiplicative TDVS instead scales semantic relevance directly by freshness and discounts stale items more aggressively, making it more suitable for freshness-sensitive RAG, temporal question answering, and statedependent episodic memories [21, 32, 39]. Thus, the additive form models a soft preference for recency, whereas the multiplicative form imposes a tighter coupling between semantic relevance and temporal freshness.

Definition 1 (Time-Decay Function). A time-decay function is a mapping 𝑤 : R ≥0 → (0, 1] that assigns a freshness weight to a data vector according to its age Δ, with 𝑤 (0) = 1 and 𝑤 (Δ) monotonically non-increasing in Δ. We use the exponential decay function 𝑤 𝜆 (Δ) = 𝑒 −𝜆Δ ,

𝜆 ≥ 0,

where 𝜆 controls the decay rate. The exponential family is well motivated in both theory and practice. It is commonly used to model the Ebbinghaus Forgetting Curve [29] and admits the interpretable half-life parameterization 𝑤ℎ (Δ) = 𝑒 − (ln 2/ℎ)Δ = 2−Δ/ℎ ,

ℎ > 0,

which is equivalent to 𝑤 𝜆 (Δ) when 𝜆 = ln 2/ℎ [43]. Similar decay functions are used for recency modeling in Generative Agents and OpenClaw [32, 34], although typically as post-retrieval scoring, or reranking components, rather than as a native search objective. Definition 2 (Time-Decayed Vector Search). Given a timestamped vector dataset D = {(𝑥𝑖 , 𝑡𝑖 )}𝑛𝑖=1 , a query vector 𝑞, a query time 𝜏, a similarity function 𝑠 (·, ·), and a time-decay function 𝑤 (·), the time-decayed vector search (TDVS) problem retrieves the top-𝑘 items under a joint semantic-temporal score 𝐹 (𝑠 (𝑞, 𝑥𝑖 ), 𝑤 (Δ𝑖 (𝜏))) , where Δ𝑖 (𝜏) = 𝜏 − 𝑡𝑖 denotes the age of item 𝑖. Formally, N𝑘tdvs (𝑞, 𝜏) = TopK𝑘(𝑥𝑖 ,𝑡𝑖 ) ∈ D 𝐹 (𝑠 (𝑞, 𝑥𝑖 ), 𝑤 (Δ𝑖 (𝜏))) . In this paper, we study two fundamental and practically important formulations of TDVS, additive and multiplicative time-decayed vector search. These two formulations capture distinct ways in which temporal freshness interacts with semantic similarity.

1When the semantic score is signed, multiplicative TDVS attenuates its magnitude

toward the neutral value zero while preserving its sign. This behavior is natural because positive and negative scores represent opposite but meaningful semantic associations: as an item ages, the strength of either association should diminish, while its direction should remain unchanged. Moreover, whenever a query has at least 𝑘 positive-score items, negative-score items cannot appear in its exact TDVS top-𝑘 result, because the strictly positive decay factor preserves score signs.

Definition 3 (Additive Time-Decayed Vector Search). The additive form of TDVS is defined by the scoring function 𝐹 add (𝑞, 𝑥𝑖 , 𝜏) = 𝛼 𝑠 (𝑞, 𝑥𝑖 ) + (1 − 𝛼) 𝑤 (Δ𝑖 (𝜏)),

𝛼 ∈ [0, 1] . 3

2.3

STR: Exact TDVS-to-MIPS Reduction

Step 3: Vector mapping. We now combine the encoded semantic features from Step 1 with the temporal factors from Step 2. For each TDVS mode 𝑚 ∈ {mul, add}, STR defines a query mapping Φ𝑄,𝑚 and a data mapping Φ𝑋 ,𝑚 . Applying these mappings to a query (𝑞, 𝜏) and a timestamped data vector (𝑥𝑖 , 𝑡𝑖 ) produces the corresponding STR query representation 𝑞e𝑚 𝜏 = Φ𝑄,𝑚 (𝑞, 𝜏) and data representation 𝑥e𝑖𝑚 = Φ𝑋 ,𝑚 (𝑥𝑖 , 𝑡𝑖 ). Since the differences among inner product, cosine similarity, and ranking-equivalent transformation of squared Euclidean distance have already been captured by the semantic encoders 𝜓𝑄 and 𝜓𝑋 , the query and data mappings apply uniformly to all three semantic measures. For multiplicative TDVS, the mapping is defined as:

The TDVS formulation in Section 2.2 is defined using a general semantic similarity function 𝑠 (·, ·). Here, a semantic measure refers to the underlying query–data measure used in vector search to quantify semantic similarity; common choices include inner product, cosine similarity, and Euclidean distance. Rather than developing a separate TDVS solution for each semantic measure, we investigate whether their additive and multiplicative TDVS objectives can be reduced to a common computational primitive. Through mathematical derivation, we show that for each of these three semantic measures, the additive and multiplicative TDVS score functions defined in Definitions 3 and 4 can be represented exactly as finite-dimensional inner products. Consequently, TDVS can be reduced to the Maximum Inner Product Search (MIPS) problem while preserving all query–data scores. We call this exact TDVS-toMIPS reduction Score-Preserving Temporal Reduction (STR). STR proceeds in three steps: semantic encoding, temporal factorization, and vector mapping. Semantic encoding converts query and data vectors so that their inner product reproduces the original semantic score. Temporal factorization decomposes exponential decay into query- and data-side factors. Vector mapping then incorporates these temporal factors into the encoded semantic features, producing mode-specific query and data vectors whose inner product equals the complete TDVS score.

Φ𝑄,mul (𝑞, 𝜏) = 𝑔𝜏(𝑇 ) 𝜓𝑄 (𝑞),

By Equations (1) and (4), the resulting representations satisfy 𝑞e𝜏mul, 𝑥e𝑖mul = 𝐹 mul (𝑞, 𝑥𝑖 , 𝜏),

 2 𝑥𝑖   𝜌    1 𝜓𝑋 (𝑥𝑖 ) =  .  2  2 − 𝜌 2 ∥𝑥𝑖 ∥ 2   

(1)

(2)

3

Step 2: Temporal factorization. Let 𝑇 be a temporal anchor; define a data-side temporal factor 𝑧𝑖(𝑇 ) , and a query-side temporal factor 𝑔𝜏(𝑇 ) as 𝑧𝑖(𝑇 ) = 𝑒 −𝜆 (𝑇 −𝑡𝑖 ) ,

𝑔𝜏(𝑇 ) = 𝑒 −𝜆 (𝜏 −𝑇 ) .

(7)

CHRONOS FRAMEWORK

To overcome the structural constraints of STR, we propose Chronos, a TDVS-native framework that removes the temporal anchor and decouples exact query–data score preservation from data–data geometry, enabling a better balance between temporal freshness and semantic similarity during indexing. Figure 2 contrasts the two solution pipelines for TDVS. STR exactly reduces TDVS to MIPS and reuses state-of-the-art MIPS indexes, whereas Chronos provides a TDVS-native framework instantiated by TANGO. Both pipelines preserve the exact TDVS objective, but differ in whether the indexing geometry is inherited from generic MIPS methods or designed specifically for TDVS.

(3)

Their product recovers the original decay exactly: 𝑔𝜏(𝑇 ) 𝑧𝑖(𝑇 ) = 𝑒 −𝜆 (𝜏 −𝑡𝑖 ) .

𝑞e𝜏add, 𝑥e𝑖add = 𝐹 add (𝑞, 𝑥𝑖 , 𝜏).

Equation (7) shows that the inner product between the mapped query and data representations is numerically identical to the target TDVS score. Consequently, STR introduces no score approximation and preserves the exact TDVS ranking. STR provides a generic and practical solution to TDVS: with only lightweight query and data transformations, it enables existing MIPS indexes to directly support TDVS without changing their search objectives. We instantiate STR with representative graphbased MIPS indexes [5, 6, 24, 28, 46], which serve as competitive baselines in our experimental evaluation (Section 5). Despite its exactness and practical convenience, STR inherits structural constraints from temporal factorization. At the representation level, transformed vectors depend on the global anchor 𝑇 . Keeping 𝑇 fixed causes newly inserted vectors with 𝑡𝑖 > 𝑇 to satisfy 𝑧𝑖(𝑇 ) > 1, progressively widening the range of transformed norms; updating 𝑇 , however, changes existing representations and may require re-encoding the stored vectors and rebuilding the index. At the geometry level, the pairwise temporal affinity 𝑧𝑖(𝑇 ) 𝑧 𝑗(𝑇 ) = 𝑒 −𝜆 (2𝑇 − (𝑡𝑖 +𝑡 𝑗 ) ) depends on the vectors’ absolute freshness relative to 𝑇 . This can make recent vectors more likely to become central hubs in the graph, which can distort semantic neighborhoods and make graph navigation more challenging.

For inner-product semantics, no additional encoding is required: 𝜓𝑄 (𝑞) = 𝑞 and 𝜓𝑋 (𝑥𝑖 ) = 𝑥𝑖 . For cosine similarity over nonzero vectors, the encoded features are their L2-normalized representations: 𝑞 𝜓𝑄 (𝑞) = ∥𝑞 ∥ 2 ,𝜓𝑋 (𝑥𝑖 ) = ∥𝑥𝑥𝑖𝑖∥ 2 . For Euclidean semantics, let 𝜌 be a fixed normalization scale such that ∥𝑞 − 𝑥𝑖 ∥ 2 ≤ 𝜌 for every valid query–data pair. We use the following ranking-equivalent transformation of squared Euclidean distance: 𝑠 L2 (𝑞, 𝑥𝑖 ) = 1 − 𝜌22 ∥𝑞 − 𝑥𝑖 ∥ 22 , which lies in [−1, 1] and maintains the same ranking as the Euclidean distance. Its semantic encoding is 2     𝜌𝑞   2 2 𝜓𝑄 (𝑞) = 1 − 𝜌 2 ∥𝑞∥ 2  ,     1  

(5)

For additive TDVS, the mapping is defined as:  √  √  𝛼 𝜓𝑄 (𝑞) 𝛼 𝜓𝑋 (𝑥𝑖 ) √ √ Φ𝑄,add (𝑞, 𝜏) = , Φ𝑋 ,add (𝑥𝑖 , 𝑡𝑖 ) = . (6) 1 − 𝛼 𝑔𝜏(𝑇 ) 1 − 𝛼 𝑧𝑖(𝑇 )

Step 1: Semantic encoding. For each semantic measure, we define a query encoder 𝜓𝑄 and a data encoder 𝜓𝑋 . Given a query 𝑞 and a data vector 𝑥𝑖 , these encoders produce semantic feature vectors 𝜓𝑄 (𝑞) and 𝜓𝑋 (𝑥𝑖 ) such that 𝑠 (𝑞, 𝑥𝑖 ) = 𝜓𝑄 (𝑞),𝜓𝑋 (𝑥𝑖 ) .

Φ𝑋 ,mul (𝑥𝑖 , 𝑡𝑖 ) = 𝑧𝑖(𝑇 ) 𝜓𝑋 (𝑥𝑖 ).

(4)

Although 𝑇 can be chosen arbitrarily, a common choice at index construction time is 𝑇 = max𝑖 𝑡𝑖 , which ensures 0 < 𝑧𝑖(𝑇 ) ≤ 1 for every data vector in the initial dataset. 4

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them through |𝑡𝑖 −𝑡 𝑗 |, eliminating the anchor from the pairwise temporal geometry. To obtain such a temporal affinity while preserving the original query–data decay, Chronos presents the Anchor-Free Temporal Kernel.

Chronos Overview. Chronos is a TDVS-native framework that preserves exact query–data scores, while providing controllable data–data geometry for efficient indexing. It has three components: an anchor-free temporal representation that recovers exponential decay from relative timestamps (Section 3.1); an exact metricization that maps queries and data to constant-norm, score-preserving representations, converting TDVS ranking into nearest-neighbor search (Section 3.2); and a controllable geometry adjustment that balances temporal freshness and semantic similarity without altering query–data scores (Section 3.3). Chronos realizes these components efficiently by evaluating query–data and data–data distances directly from the original embeddings and timestamps, without materializing the mapped representations (Section 3.4).

Definition 5 (Anchor-Free Temporal Kernel). For timestamps 𝑠, 𝑡 ∈ R, Chronos defines the anchor-free temporal kernel as 𝑘𝜆 (𝑠, 𝑡) = 𝑒 −𝜆 |𝑠 −𝑡 | ,

𝜆 ≥ 0.

(8)

This kernel is the standard one-dimensional Laplacian kernel applied to timestamps. For every valid query–data pair 𝜏 ≥ 𝑡𝑖 , the kernel exactly recovers the original temporal decay: 𝑘𝜆 (𝜏, 𝑡𝑖 ) = 𝑒 −𝜆 |𝜏 −𝑡𝑖 | = 𝑒 −𝜆 (𝜏 −𝑡𝑖 ) . Moreover, the kernel is symmetric, 𝑘𝜆 (𝑠, 𝑡) = 𝑘𝜆 (𝑡, 𝑠), and depends only on the relative temporal separation |𝑠 −𝑡 |. It is also translation invariant: 𝑘𝜆 (𝑠 +𝑐, 𝑡 +𝑐) = 𝑘𝜆 (𝑠, 𝑡), 𝑐 ∈ R. Hence, shifting the time origin, or inserting data with later timestamps does not alter the temporal affinities among indexed vectors. To use this temporal kernel as a geometric building block for TDVS, we establish its geometric realization in a Hilbert space: Lemma 1 (below) shows that there exists a unit-norm feature map whose pairwise inner products exactly recover 𝑘𝜆 . This representation allows the temporal component to be combined with unitnorm semantic embeddings while preserving the TDVS score and, as shown in Section 3.2, enables its exact metricization.

Embedding regime. Modern embedding models and retrieval pipelines increasingly use normalized representations, projecting embeddings onto the unit sphere at model output, or before indexing [31, 41, 57]. Normalization removes vector magnitude as an additional signal and unifies common similarity measures. For unitnorm embeddings, ⟨𝑞, 𝑥𝑖 ⟩ = cos(𝑞, 𝑥𝑖 ) = 1 − 21 ∥𝑞 − 𝑥𝑖 ∥ 22 . Thus, inner product, cosine similarity, and canonical unit-sphere Euclidean similarity2 induce identical TDVS objectives. Chronos targets this important regime and provides a unified metricization framework. Chronos can be extended to non-normalized embeddings through norm completion, which introduces an additional norm-dependent geometric component orthogonal to the temporal design studied here. We leave this extension to future work.

3.1

Chronos

Lemma 1 (Unit-Norm Hilbert embedding of the temporal kernel). For every 𝜆 ≥ 0, the anchor-free temporal kernel 𝑘𝜆 is positive semidefinite. Consequently, there exists a Hilbert space H𝜆 , together with a feature map 𝜙 𝜆 : R → H𝜆 , such that for all 𝑠, 𝑡 ∈ R,

Anchor-Free Temporal Representation

⟨𝜙 𝜆 (𝑠), 𝜙 𝜆 (𝑡)⟩ = 𝑘𝜆 (𝑠, 𝑡),

TDVS directly specifies only the temporal decay for a valid query– data pair, namely 𝑒 −𝜆 (𝜏 −𝑡𝑖 ) for 𝜏 ≥ 𝑡𝑖 . However, index construction must also compare two data vectors and therefore requires a symmetric pairwise temporal affinity between their timestamps 𝑡𝑖 and 𝑡 𝑗 . STR implicitly induces an anchor-dependent pairwise temporal affinity 𝑒 −𝜆 (2𝑇 − (𝑡𝑖 +𝑡 𝑗 ) ) , which measures the joint freshness of the two data vectors relative to 𝑇 . Chronos introduces an anchorfree form 𝑒 −𝜆 |𝑡𝑖 −𝑡 𝑗 | , which depends only on their relative temporal separation. Figure 3 illustrates this distinction. STR relates both timestamps to the global anchor 𝑇 , whereas Chronos directly relates

∥𝜙 𝜆 (𝑠)∥ = ∥𝜙 𝜆 (𝑡)∥ = 1.

(9)

Proof. For 𝜆 > 0, the one-dimensional ∫ ∞ Laplacian kernel has the spectral representation 𝑒 −𝜆 |𝑠 −𝑡 | = −∞ 𝑒 i𝜔 (𝑠 −𝑡 ) 𝜋 (𝜆2𝜆+𝜔 2 ) 𝑑𝜔. Its spectral density 𝜆/(𝜋 (𝜆 2 + 𝜔 2 )) is nonnegative. By Bochner’s theorem, 𝑘𝜆 is positive semidefinite. When 𝜆 = 0, the kernel reduces to the constant kernel 𝑘 0 (𝑠, 𝑡) = 1, which is also positive semidefinite. Positive semidefiniteness guarantees the existence of the Hilbert space H𝜆 and the associated feature map 𝜙 𝜆 . Moreover, for every 𝑡 ∈ R, ∥𝜙 𝜆 (𝑡)∥ 2 = 𝑘𝜆 (𝑡, 𝑡) = 1, hence ∥𝜙 𝜆 (𝑡)∥ = 1. □ The feature map 𝜙 𝜆 is used only for theoretical construction and need not be explicitly constructed or stored; all required inner products can be evaluated exactly using Equation (8). Section 3.2 builds on this representation to derive the exact TDVS metricization.

2 Euclidean retrieval minimizes ∥𝑞 − 𝑥 ∥ . STR uses the ranking-equivalent similarity 𝑖 2 1 − 2∥𝑞 − 𝑥𝑖 ∥ 22 /𝜌 2 . Under Chronos’s unit-norm embedding regime, 𝜌 = 2, and this score reduces to 1 − ∥𝑞 − 𝑥𝑖 ∥ 22 /2 = ⟨𝑞, 𝑥𝑖 ⟩ .

5

3.2

Exact TDVS Metricization

Base data geometry. Beyond guaranteeing exact query–data score preservation, the metricization also determines the pairwise distances among indexed data vectors, thereby defining the base data geometry used for index construction. For two data vectors (𝑥𝑖 , 𝑡𝑖 ) and (𝑥 𝑗 , 𝑡 𝑗 ), let 𝑐𝑖 𝑗 = ⟨𝑥𝑖 , 𝑥 𝑗 ⟩ denote their semantic similarity and let 𝑘𝑖 𝑗 = 𝑘𝜆 (𝑡𝑖 , 𝑡 𝑗 ) = 𝑒 −𝜆 |𝑡𝑖 −𝑡 𝑗 | denote their temporal affinity. For multiplicative TDVS, the squared distance between the mapped data representations in the corresponding Hilbert space is

By Lemma 1, each timestamp 𝑡 has a unit-norm temporal feature 𝜙 𝜆 (𝑡), while the semantic embeddings used by Chronos are also unit-norm. In this section, for each TDVS mode, Chronos combines the unit-norm semantic embeddings with the unit-norm temporal features to map queries and data vectors into a mode-specific Hilbert space. The resulting mappings preserve the complete TDVS score and assign unit norm to every mapped query and data vector. These properties are formalized in Theorem 1 (below), which shows that the mapped inner product exactly recovers the TDVS score and that score maximization is therefore equivalent to nearest-neighbor search under the induced Hilbert-space metric. We refer to this exact equivalence as TDVS metricization. For multiplicative TDVS, the target score is the product of semantic similarity and temporal freshness. Chronos combines each semantic embedding with its corresponding temporal feature using the tensor product: 𝑄 mul (𝑞, 𝜏) = 𝑞 ⊗ 𝜙 𝜆 (𝜏),

𝑃mul (𝑥𝑖 , 𝑡𝑖 ) = 𝑥𝑖 ⊗ 𝜙 𝜆 (𝑡𝑖 ),

base 𝐷 mul (𝑥𝑖 , 𝑥 𝑗 ) = 𝑃mul (𝑥𝑖 , 𝑡𝑖 ) − 𝑃 mul (𝑥 𝑗 , 𝑡 𝑗 )

2

= 2(1 − 𝑘𝑖 𝑗 ) + 2𝑘𝑖 𝑗 (1 − 𝑐𝑖 𝑗 ).

(13)

For additive TDVS, the corresponding squared distance is base 𝐷 add (𝑥𝑖 , 𝑥 𝑗 ) = 𝑃add (𝑥𝑖 , 𝑡𝑖 ) − 𝑃 add (𝑥 𝑗 , 𝑡 𝑗 )

2

= 2(1 − 𝛼)(1 − 𝑘𝑖 𝑗 ) + 2𝛼 (1 − 𝑐𝑖 𝑗 ).

(14)

These expressions show how the base geometry combines semantic similarity and temporal affinity under the two TDVS modes. In the additive case, the two contributions are explicitly weighted by 𝛼 and 1−𝛼. In the multiplicative case, the semantic component 2𝑘𝑖 𝑗 (1−𝑐𝑖 𝑗 ) is coupled to the temporal affinity 𝑘𝑖 𝑗 . As the temporal separation |𝑡𝑖 − 𝑡 𝑗 | increases, 𝑘𝑖 𝑗 decreases, gradually reducing the influence of semantic similarity. When 𝑘𝑖 𝑗 approaches zero, the squared distance approaches 2 regardless of 𝑐𝑖 𝑗 , weakening the connections between semantically related vectors with distant timestamps. Thus, the base multiplicative geometry naturally favors temporal locality, but does not provide an independent mechanism for controlling cross-time semantic connectivity. Section 3.3 introduces a score-preserving geometry adjustment that provides such control without changing any query–data TDVS score.

(10)

where both lie in the tensor-product space Hmul = R𝑑 ⊗ H𝜆 . For additive TDVS, the target score is the weighted sum of semantic similarity and temporal freshness. Chronos combines each semantic embedding with its corresponding temporal feature using a weighted direct sum:     √ √ 𝛼𝑞 𝛼 𝑥𝑖 𝑄 add (𝑞, 𝜏) = √ , 𝑃add (𝑥𝑖 , 𝑡𝑖 ) = √ . (11) 1 − 𝛼 𝜙 𝜆 (𝜏) 1 − 𝛼 𝜙 𝜆 (𝑡𝑖 ) Both representations lie in the direct-sum space Hadd = R𝑑 ⊕ H𝜆 . The square-root scaling ensures that their inner product reproduces the weights 𝛼 and 1 − 𝛼 in the additive TDVS objective. In the remainder of this section, all inner products, norms, and distances are taken in the mode-specific Hilbert space H𝑚 . We use 2 to denote its 𝑑𝑚 to denote the Hilbert-space metric and 𝐷𝑚 = 𝑑𝑚 squared form. Since squaring preserves nonnegative order, 𝑑𝑚 and 𝐷𝑚 induce the same nearest-neighbor ordering.

3.3

Query-Orthogonal TimeLift

To independently adjust the influence of semantic similarity on data–data geometry without changing the TDVS objective, Chronos introduces a query-orthogonal geometry adjustment called TimeLift. The key idea is to append an additional semantic component to each data representation, while appending a zero component to every query representation. The added component therefore affects data–data distances but contributes nothing to query–data scores. Theorem 2 (below) formalizes the key query-equivalence property of TimeLift: for any fixed 𝜅, it changes the data–data geometry while preserving all query–data TDVS scores and rankings. Let H𝑚 denote the mode-specific Hilbert space defined in Section 3.2. For a geometry parameter 𝜅 ≥ 0, TimeLift extends this b𝑚 = H𝑚 ⊕ R𝑑 . space to H

Theorem 1 (Exact TDVS Metricization). For each mode 𝑚 ∈ {mul, add} and every valid query–data pair 𝜏 ≥ 𝑡𝑖 , the Chronos mappings satisfy ⟨𝑄𝑚 (𝑞, 𝜏), 𝑃𝑚 (𝑥𝑖 , 𝑡𝑖 )⟩ = 𝐹𝑚 (𝑞, 𝑥𝑖 , 𝜏) and ∥𝑄𝑚 (𝑞, 𝜏)∥ = ∥𝑃𝑚 (𝑥𝑖 , 𝑡𝑖 ) ∥ = 1. TDVS score maximization is equivalent to minimizing either the Hilbert-space metric 𝑑𝑚 or its squared form 𝐷𝑚 . Proof. For multiplicative TDVS, applying the tensor-product identity and Equation (9) gives ⟨𝑄 mul, 𝑃mul ⟩ = ⟨𝑞, 𝑥𝑖 ⟩ ⟨𝜙 𝜆 (𝜏), 𝜙 𝜆 (𝑡𝑖 )⟩ = ⟨𝑞, 𝑥𝑖 ⟩𝑒 −𝜆 (𝜏 −𝑡𝑖 ) = 𝐹 mul (𝑞, 𝑥𝑖 , 𝜏). For additive TDVS, ⟨𝑄 add, 𝑃add ⟩ = 𝛼 ⟨𝑞, 𝑥𝑖 ⟩ + (1 − 𝛼) ⟨𝜙 𝜆 (𝜏), 𝜙 𝜆 (𝑡𝑖 )⟩ = 𝛼 ⟨𝑞, 𝑥𝑖 ⟩ + (1 − 𝛼)𝑒 −𝜆 (𝜏 −𝑡𝑖 ) = 𝐹 add (𝑞, 𝑥𝑖 , 𝜏). Since the semantic embeddings and temporal features are unit-norm, the tensor-product norm gives ∥𝑄 mul ∥ = ∥𝑃mul ∥ = 1, while the direct-sum construction gives ∥𝑄 add ∥ 2 = ∥𝑃add ∥ 2 = 𝛼 + (1 − 𝛼) = 1. Finally, expanding the squared distance for either mode gives

Definition 6 (Query-Orthogonal TimeLift). For each TDVS mode 𝑚 ∈ {mul, add}, TimeLift maps a query (𝑞, 𝜏) and a timestamped data vector (𝑥𝑖 , 𝑡𝑖 ) to     𝑄𝑚 (𝑞, 𝜏) 𝑃𝑚 (𝑥𝑖 , 𝑡𝑖 ) (𝜅 ) (𝜅 ) b b √ 𝑄𝑚 (𝑞, 𝜏) = , 𝑃𝑚 (𝑥𝑖 , 𝑡𝑖 ) = , (15) 0𝑑 𝜅 𝑥𝑖 where 0𝑑 is the zero vector in R𝑑 .

2

√ The appended semantic component 𝜅𝑥𝑖 is query-orthogonal (𝜅 ) b𝑚 because the corresponding component of 𝑄 (𝑞, 𝜏) is 0𝑑 . Let 𝑑𝑚(𝜅 ) (𝜅 ) (𝜅 ) 2 be the TimeLift metric and 𝐷𝑚 = (𝑑𝑚 ) be its squared form.

2

𝐷𝑚 (𝑞, 𝑥𝑖 ) = (𝑑𝑚 (𝑞, 𝑥𝑖 )) = ∥𝑄𝑚 − 𝑃𝑚 ∥ = 2 − 2𝐹𝑚 (𝑞, 𝑥𝑖 , 𝜏). (12) Thus, minimizing the Hilbert-space distance 𝑑𝑚 or its squared form 𝐷𝑚 is exactly equivalent to maximizing the TDVS score. □ 6

query

Theorem 2 (Query-Eqivalent TimeLift). For every 𝜅 ≥ 0 and each TDVS mode 𝑚 ∈ {mul, add}, TimeLift preserves all query–data TDVS scores. Moreover, every mapped query has √ unit norm, whereas all mapped data vectors have the same norm 1 + 𝜅. For any fixed 𝜅, nearest-neighbor search using either 𝑑𝑚(𝜅 ) or its squared form 𝐷𝑚(𝜅 ) preserves the exact TDVS ranking.

TimeLift ��

Layer � − � TimeLift ��−�

(16) Layer �

Since ∥𝑄𝑚√∥ = ∥𝑃𝑚 ∥ = ∥𝑥𝑖 ∥ = 1, the mapped query and data norms are 1 and 1 + 𝜅, respectively. Hence, 2  2 (𝜅 ) b𝑚 − 𝑃b𝑚(𝜅 ) = 2+𝜅 −2𝐹𝑚 (𝑞, 𝑥𝑖 , 𝜏). 𝐷𝑚(𝜅 ) (𝑞, 𝑥𝑖 ) = 𝑑𝑚(𝜅 ) (𝑞, 𝑥𝑖 ) = 𝑄 (17) For any fixed 𝜅, minimizing either 𝑑𝑚(𝜅 ) or its squared form 𝐷𝑚(𝜅 ) is exactly equivalent to maximizing the TDVS score. □

TimeLift ��

(18)

For additive TDVS, the corresponding squared distance is (𝜅 ) (𝜅 ) (𝜅 ) 𝐷 add (𝑥𝑖 , 𝑥 𝑗 ) = 𝑃badd (𝑥𝑖 , 𝑡𝑖 ) − 𝑃badd (𝑥 𝑗 , 𝑡 𝑗 )

2

= 2(1 − 𝛼)(1 − 𝑘𝑖 𝑗 ) + 2(𝛼 + 𝜅)(1 − 𝑐𝑖 𝑗 ).

(19)

Compared with the base data geometry, TimeLift introduces a timestamp-independent semantic term 2𝜅 (1 − 𝑐𝑖 𝑗 ). In the multiplicative case, the coefficient of semantic separation increases from 𝑘𝑖 𝑗 to 𝑘𝑖 𝑗 +𝜅. Thus, even when two timestamps are far apart and 𝑘𝑖 𝑗 approaches zero, their squared distance still depends on semantic similarity through 2𝜅 (1 − 𝑐𝑖 𝑗 ). In the additive case, TimeLift similarly adjusts the semantic contribution from 𝛼 to 𝛼 + 𝜅 in the data geometry, without changing the weight 𝛼 in the TDVS objective. The parameter 𝜅 controls the indexing geometry: 𝜅 = 0 recovers the base geometry, while larger values strengthen the influence of semantic similarity and connectivity across distant timestamps. Because all 𝜅 values preserve the exact TDVS ranking while inducing distinct data–data geometries, Chronos defines a query-equivalent but geometry-distinct family. However, when constructing an index, using one global 𝜅 forces a trade-off between temporal locality and cross-time semantic connectivity. TANGO (Section 4) addresses this trade-off by assigning layer-specific 𝜅 ℓ values within a hierarchical graph while preserving the same TDVS ordering across all layers.

3.4

Base layer: smaller ��

stronger temporal locality

index construction (data–data) in the mapped Hilbert spaces, allowing the score-preserving and geometry-controllable Hilbert-space geometry to be realized directly from the original embeddings and timestamps without materializing the mapped representations. Specifically, during query processing, Chronos directly evaluates the original TDVS score 𝐹𝑚 (𝑞, 𝑥𝑖 , 𝜏). Equivalently, for any fixed 𝜅, the same candidate ordering can be expressed through the query– data squared distance 𝐷𝑚(𝜅 ) (𝑞, 𝑥𝑖 ) in Equation (17). During index construction, Chronos computes data–data squared distances using (𝜅 ) (𝜅 ) 𝐷 mul (𝑥𝑖 , 𝑥 𝑗 ) and 𝐷 add (𝑥𝑖 , 𝑥 𝑗 ) in Equations (18)–(19). The TANGO graph index introduced in Section 4 uses these data–data squared distances to construct its graph neighborhoods, while using the exact TDVS score to guide query traversal.

2

= 2(1 − 𝑘𝑖 𝑗 ) + 2(𝑘𝑖 𝑗 + 𝜅)(1 − 𝑐𝑖 𝑗 ).

same TDVS objective �� at all layers

Figure 4: TANGO Hierarchical Index.

Controllable data geometry. Although 𝜅 does not affect any query– data score, it changes the pairwise distances among indexed data vectors. Using 𝑐𝑖 𝑗 and 𝑘𝑖 𝑗 defined in Section 3.2, the squared TimeLift distance for multiplicative TDVS is (𝜅 ) (𝜅 ) (𝜅 ) 𝐷 mul (𝑥𝑖 , 𝑥 𝑗 ) = 𝑃bmul (𝑥𝑖 , 𝑡𝑖 ) − 𝑃bmul (𝑥 𝑗 , 𝑡 𝑗 )

Upper layers: larger �� stronger cross-time semantic connectivity

...

Proof. By Definition 6 and Theorem 1, D E √ (𝜅 ) b(𝜅 ) b𝑚 𝑄 , 𝑃𝑚 = ⟨𝑄𝑚 , 𝑃𝑚 ⟩ + 0𝑑 , 𝜅𝑥𝑖 = 𝐹𝑚 (𝑞, 𝑥𝑖 , 𝜏).

Layer �

4

TANGO: A TDVS-NATIVE GRAPH INDEX

Overview. Chronos exposes a family of query-equivalent but geometry-distinct metric spaces parameterized by 𝜅. However, a single value of 𝜅 defines a fixed geometry that is difficult to simultaneously support long-range navigation and local candidate refinement. In this section, we propose TANGO, short for Time-Aware Navigable Graph with Query-Orthogonal TimeLift, a TDVS-native hierarchical graph index that realizes different Chronos geometries across graph layers while preserving a common query objective. TANGO follows the hierarchical organization of HNSW [25]. As illustrated in Figure 4, TANGO’s key distinction from HNSW is that each layer ℓ is constructed under a distinct TimeLift geometry parameterized by 𝜅 ℓ : larger values in upper layers strengthen crosstime semantic connectivity, while the smaller 𝜅 0 at the base layer preserves temporal locality. This layer specialization allows TANGO to navigate rapidly across temporally distant regions in the upper layers and then quickly focus on locally relevant semantic–temporal neighborhoods at the base layer. Importantly, layer-specific geometries introduce no objective mismatch: 𝜅 ℓ affects only the data–data geometry used for graph construction, while query–data ordering remains exactly the TDVS ordering. Consequently, all layers can be searched using the same exact TDVS score 𝐹𝑚 (𝑞, 𝑥𝑖 , 𝜏) within a single top-down traversal. Moreover, with temporal factors cached and shared across layers, TANGO avoids repeated exponential computations for temporal decay and requires only one semantic inner

Efficient Realization

Chronos uses the Hilbert-space mappings only for theoretical construction and does not explicitly construct or store the temporal features 𝜙 𝜆 (𝑡), the tensor-product or direct-sum representations (𝜅 ) b𝑚 𝑄𝑚 (𝑞, 𝜏) and 𝑃𝑚 (𝑥𝑖 , 𝑡𝑖 ), or their TimeLift extensions 𝑄 (𝑞, 𝜏) and (𝜅 ) 𝑃b𝑚 (𝑥𝑖 , 𝑡𝑖 ). This is because Chronos derives exact formulas for the pairwise distance required by query processing (query–data) and 7

Layer-shared temporal factor caching. During index construction and query traversal, TANGO would repeatedly compute the data– data temporal affinity 𝑒 −𝜆 |𝑡𝑖 −𝑡 𝑗 | and the query–data decay 𝑒 −𝜆 (𝜏 −𝑡𝑖 ) , causing nontrivial computational overhead. Since these terms depend only on timestamps and the fixed decay rate 𝜆, they can be factorized into per-data and per-query temporal factors and cached for reuse. Moreover, these cached factors are independent of the layer-specific parameter 𝜅 ℓ , allowing the same values to be shared across all graph layers. Let 𝑡 0 be a reference timestamp chosen for numerical stability; TANGO sets it to the latest timestamp when the index is initialized. It caches one per-data temporal factor 𝑏𝑖 for each indexed vector and computes one per-query temporal factor 𝑎𝜏 for each query:

Algorithm 1: TANGO Layer Search Input: Layer Gℓ , TDVS mode 𝑚 ∈ {mul, add} , search budget 𝑒 𝑓 , cached data factors {𝑏𝑖 } , phase 𝑟 ∈ {build, query} , entry set E ; vector to be inserted 𝑣 = (𝑥 𝑣 , 𝑡 𝑣 ) and layer parameter 𝜅 ℓ if 𝑟 = build; query (𝑞, 𝜏 ) and query factor 𝑎𝜏 if

𝑟 = query Output: Candidate set W 1 if 𝑟 = build then (𝜅 )

For any vertex 𝑢 ∈ Vℓ , define the smaller-is-better key 𝛿𝑟 (𝑢 ) ← 𝐷𝑚 ℓ (𝑥 𝑣 , 𝑥𝑢 ) ,

2

min{𝑏 ,𝑏 } where 𝑘 𝑣𝑢 ← max{𝑏𝑣 ,𝑏𝑢 } ; 𝑣 𝑢 3 4

else For any vertex 𝑢 ∈ Vℓ , define the smaller-is-better key 𝛿𝑟 (𝑢 ) ← −𝐹𝑚 (𝑞, 𝑥𝑢 , 𝜏 ) , where 𝑒 −𝜆 (𝜏 −𝑡𝑢 ) ← 𝑎𝜏 𝑏𝑢 ;

Initialize C (a min-priority queue of unexpanded candidates) and W (a max-priority queue of the current best vertices) with E , both ordered by 𝛿𝑟 ; set the visited set Z ← E; 6 while C ≠ ∅ do 7 𝑐 ← remove the vertex with the smallest 𝛿𝑟 from C ; 8 𝑤 ← the vertex with the largest 𝛿𝑟 in W ; 9 if | W | ≥ 𝑒 𝑓 and 𝛿𝑟 (𝑐 ) > 𝛿𝑟 (𝑤 ) then 10 break; 5

Z ← Z ∪ {𝑢 } ; Compute 𝛿𝑟 (𝑢 ) using the phase-specific key; if | W | < 𝑒 𝑓 or 𝛿𝑟 (𝑢 ) < max𝑧 ∈W 𝛿𝑟 (𝑧 ) then Insert 𝑢 into C and W ; if | W | > 𝑒 𝑓 then Remove the largest-𝛿𝑟 vertex from W ;

14 15 16 17 18

𝑒 −𝜆 (𝜏 −𝑡𝑖 ) = 𝑎𝜏 𝑏𝑖 ,

product and constant-time scalar operations per visited candidate, enabling efficient query processing.

Layer-Wise Graph Geometry

Let G = {Gℓ = (Vℓ , Eℓ )}𝐿ℓ=0 denote the TANGO hierarchy, where ℓ = 0 is the base layer and Vℓ+1 ⊆ Vℓ . As in HNSW, each data vector is assigned a maximum level, and higher layers contain progressively fewer vectors. TANGO assigns a separate nonnegative TimeLift parameter to each graph layer: 𝜿 = (𝜅 0, 𝜅 1, . . . , 𝜅𝐿 ),

(21)

𝜅 ℓ ≥ 0.

𝑒 −𝜆 |𝑡𝑖 −𝑡 𝑗 | =

min{𝑏𝑖 , 𝑏 𝑗 } . max{𝑏𝑖 , 𝑏 𝑗 }

(22)

Caching these temporal factors eliminates repeated exponentiation from pairwise comparisons: TANGO performs one exponential computation per inserted vector and one per query, while subsequent temporal terms are obtained using only scalar multiplication or division and are shared across all graph layers. Unlike the temporal anchor 𝑇 in STR, 𝑡 0 is only a numerical reference and does not alter any retrieval score, data geometry, or graph edge. As time advances, although keeping 𝑡 0 fixed does not affect correctness, the numerical scale of the cached factors may gradually drift. TANGO can then replace 𝑡 0 with a newer reference 𝑡 0′ and uniformly rescale each ′ cached factor as 𝑏𝑖 ← 𝑒 −𝜆 (𝑡0 −𝑡0 ) 𝑏𝑖 , without recomputing pairwise distances or rebuilding the graph.

return W ;

4.1

𝑎𝜏 = 𝑒 −𝜆 (𝜏 −𝑡0 ) .

The required query–data decay and data–data temporal affinity are then computed exactly as

foreach 𝑢 ∈ Nℓ (𝑐 ) do if 𝑢 ∉ Z then

11 12 13

19

𝑏𝑖 = 𝑒 𝜆 (𝑡𝑖 −𝑡0 ) ,

4.2

Index Construction and Online Insertion

TANGO retains the hierarchical organization, level assignment, and bounded-degree structure of HNSW, but refines the proximity computations used to construct the graph. Specifically, every distance-based decision at layer ℓ uses the layer-specific squared TimeLift distance 𝐷𝑚(𝜅ℓ ) (𝑥𝑖 , 𝑥 𝑗 ) (Equations (18)–(19)). The temporal affinity required by this distance is computed from the cached facmin{𝑏 ,𝑏 } tors as 𝑘𝑖 𝑗 = 𝑒 −𝜆 |𝑡𝑖 −𝑡 𝑗 | = max{𝑏𝑖𝑖 ,𝑏 𝑗𝑗 } . Consequently, TANGO retains HNSW’s efficient hierarchical navigation while natively incorporating the semantic–temporal geometry of TDVS, thereby supporting efficient index construction and query processing.

(20)

A larger 𝜅 ℓ gives semantic similarity greater influence over the data geometry of layer ℓ, whereas a smaller 𝜅 ℓ preserves stronger temporal locality. Within TANGO’s multi-layer hierarchy, sparse upper layers benefit from stronger cross-time semantic connectivity for long-range navigation, whereas lower layers benefit from greater temporal locality for fine-grained refinement. Thus, TANGO adopts a nondecreasing schedule: 0 ≤ 𝜅 0 ≤ 𝜅 1 ≤ · · · ≤ 𝜅𝐿 . From Equation (17), 𝐷𝑚(𝜅ℓ ) (𝑞, 𝑥𝑖 ) = 2 + 𝜅 ℓ − 2𝐹𝑚 (𝑞, 𝑥𝑖 , 𝜏). Hence, different values of 𝜅 ℓ add different layer-dependent constants to the squared query–data distances, so their numerical values are not directly comparable across layers. Nevertheless, the candidate ordering remains identical across all layers. TANGO therefore uses the common TDVS score 𝐹𝑚 throughout query traversal, while 𝜅 ℓ determines only the graph neighborhoods constructed at layer ℓ. At layer ℓ, all data–data comparisons are based on 𝐷𝑚(𝜅ℓ ) (𝑥𝑖 , 𝑥 𝑗 ), i.e., the multiplicative or additive TimeLift distance defined in Equations (18) and (19), respectively. Since graph construction depends only on distance ordering, the squared distance can be used without computing its square root.

TANGO layer search. Algorithm 1 presents the common layersearch procedure used during both construction and query processing. The two phases share the same best-first queue operations but use different ranking keys. During construction, vertices at layer ℓ are ordered by the layer-specific squared TimeLift distance 𝐷𝑚(𝜅ℓ ) (𝑥 𝑣 , 𝑥𝑢 ), whose temporal affinity is obtained from 𝑏 𝑣 and 𝑏𝑢 . During query processing, vertices are instead ordered by the negative exact TDVS score −𝐹𝑚 (𝑞, 𝑥𝑢 , 𝜏), with the query–data decay evaluated as 𝑎𝜏 𝑏𝑢 . The shared search procedure therefore does not impose a common proximity measure on the two phases. The construction key determines the neighborhoods and topology of each layer, whereas 8

Algorithm 2: Index Construction and Online Insertion

Algorithm 3: Query Processing

Input: TANGO hierarchy G = { Gℓ }ℓ𝐿=0 , layer parameters 𝜿 , TDVS mode 𝑚 ∈ {mul, add} , vector to be inserted 𝑣 = (𝑥 𝑣 , 𝑡 𝑣 ) , decay rate 𝜆 , reference timestamp 𝑡 0 , cached data factors {𝑏𝑖 } , budget efConstruction, and degree limits

Input: TANGO hierarchy G = { Gℓ }ℓ𝐿=0 , query (𝑞, 𝜏 ) , decay rate 𝜆 , TDVS mode 𝑚 ∈ {mul, add} , reference timestamp 𝑡 0 , cached data factors {𝑏𝑖 } , result size 𝑘 , budget efSearch ≥ 𝑘 b𝑘 Output: Approximate TDVS top-𝑘 result N 1 Compute the query factor 𝑎𝜏 ← 𝑒 −𝜆 (𝜏 −𝑡 0 ) for reuse across layers; 2 Set the entry set E ← { global entry point } ; 3 for ℓ = 𝐿 down to 1 do 4 E ← call TANGO Layer Search ( Gℓ , 𝑚, 1, {𝑏𝑖 }, query, E, (𝑞, 𝜏 ), 𝑎𝜏 ) ;

{𝑀ℓ } Output: Updated hierarchy G and cached factors {𝑏𝑖 } 1 Compute and cache 𝑏 𝑣 ← 𝑒 𝜆 (𝑡 𝑣 −𝑡 0 ) for reuse across all layers; 2 Sample the maximum level 𝐿𝑣 using the HNSW level distribution; 3 if G is empty then 4 for ℓ = 0 to 𝐿𝑣 do 5 Add 𝑣 to Vℓ ; 7

9 10

Set the entry set E ← { global entry point } ; for ℓ = 𝐿 down to 𝐿𝑣 + 1 do E ← call TANGO Layer Search ( Gℓ , 𝑚, 1, {𝑏𝑖 }, build, E, 𝑣, 𝜅 ℓ ) ;

11 12

for ℓ = min{𝐿, 𝐿𝑣 } down to 0 do Wℓ ← call TANGO Layer Search ( Gℓ , 𝑚, efConstruction, {𝑏𝑖 }, build, E, 𝑣, 𝜅 ℓ ) ;

8

14 15 16 17 18

E ← Wℓ ;

19 20 21 22

if 𝐿𝑣 > 𝐿 then for ℓ = 𝐿 + 1 to 𝐿𝑣 do Add 𝑣 to Vℓ ; Set 𝑣 as the global entry point and 𝐿 ← 𝐿𝑣 ;

23 24

return ( G, {𝑏𝑖 } ) ;

the query key directly follows the target TDVS objective. In both phases, temporal factor caching eliminates repeated exponentiation during graph traversal. Index construction. Algorithm 2 presents the complete index construction procedure. To insert a timestamped vector 𝑣 = (𝑥 𝑣 , 𝑡 𝑣 ), TANGO computes and caches its temporal factor 𝑏 𝑣 = 𝑒 𝜆 (𝑡 𝑣 −𝑡0 ) once and reuses it across all graph layers. TANGO retains HNSW’s hierarchical insertion and diversified neighbor selection mechanisms, but replaces their underlying proximity measure with the TimeLift geometry of the current layer. During the top-down traversal and candidate exploration, every data–data comparison at layer ℓ is evaluated using the layer-specific squared TimeLift distance 𝐷𝑚(𝜅ℓ ) (𝑥𝑖 , 𝑥 𝑗 ), with the temporal affinity obtained directly from the cached factors.

7

b𝑘 ; return N

𝑢 ∈W0

5

EXPERIMENTAL EVALUATION

In this section, we conduct experiments to evaluate TANGO. Our evaluation addresses three questions: (1) how TANGO compares to state-of-the-art graph-based methods in query and index performance (Section 5.2); (2) how robust TANGO is across different temporal settings (Section 5.3); and (3) how efficiently TANGO supports online insertion (Section 5.4). All experiments are conducted on a server equipped with two Intel Xeon Platinum 8458P processors, providing 88 physical cores and 881 GiB of memory. The server runs Ubuntu 22.04.5 with Linux 5.15. All methods use 64 threads for index construction and a single thread for query processing. All implementations are compiled with GCC 11.4 in C++17 release mode using -O3 and -march=native.

Online insertion. The same procedure directly supports online insertion. For each newly arriving vector, TANGO computes its cached temporal factor 𝑏 𝑣 and performs the corresponding local searches, edge insertions, and adjacency-list pruning. Since 𝐷𝑚(𝜅ℓ ) (𝑥𝑖 , 𝑥 𝑗 ) depends on the relative timestamp difference |𝑡𝑖 − 𝑡 𝑗 |, inserting a vector leaves existing data–data distances unchanged and requires no global updates to previously indexed vectors. TANGO remains temporally stable under subsequent insertions.

4.3

W0 ← call TANGO Layer Search ( G0 , 𝑚, efSearch, {𝑏𝑖 }, query, E, (𝑞, 𝜏 ), 𝑎𝜏 ) ; b𝑘 ← TopK𝑘 𝐹𝑚 (𝑞, 𝑥𝑢 , 𝜏 ) ; N

query–data distances within layer ℓ and therefore does not change candidate ordering. TANGO consequently traverses every layer using the same exact TDVS score 𝐹𝑚 (𝑞, 𝑥𝑖 , 𝜏). Algorithm 3 presents the query processing procedure. TANGO computes the query factor 𝑎𝜏 = 𝑒 −𝜆 (𝜏 −𝑡0 ) once and reuses it across graph layers during hierarchical traversal. For each visited vertex 𝑢, the query–data decay is obtained as 𝑎𝜏 𝑏𝑢 , and the smaller-isbetter key is defined as 𝛿 query (𝑢) = −𝐹𝑚 (𝑞, 𝑥𝑢 , 𝜏) in Algorithm 1. By invoking Algorithm 1, TANGO performs upper-layer navigation and base-layer exploration under the same exact TDVS scoring rule, and returns the 𝑘 candidates with the largest TDVS scores. Taken together, TANGO assigns distinct roles to graph geometry and query scoring. The layer-specific graph geometries determine the neighborhoods and connectivity available for traversal, while the exact TDVS score determines the exploration priority of visited vertices and their ranking. With temporal factor caching, TANGO supports TDVS via one hierarchical traversal without materializing Chronos representations, requiring only one semantic inner product and constant-time scalar operations per visited candidate.

(𝜅 ) Sℓ ← call Diversified Neighbor Selection (𝑣, Wℓ , 𝑀ℓ , 𝐷𝑚 ℓ (·, ·), {𝑏𝑖 } ) ; Add 𝑣 to Vℓ and set Nℓ (𝑣) ← Sℓ ; foreach 𝑢 ∈ Sℓ do Add the reverse edge Nℓ (𝑢 ) ← Nℓ (𝑢 ) ∪ {𝑣 } ; if | Nℓ (𝑢 ) | > 𝑀ℓ then Nℓ (𝑢 ) ← call Diversified Neighbor Selection (𝜅 ) (𝑢, Nℓ (𝑢 ), 𝑀ℓ , 𝐷𝑚 ℓ (·, ·), {𝑏𝑖 } ) ;

13

6

5

Set 𝑣 as the global entry point and 𝐿 ← 𝐿𝑣 ; return ( G, {𝑏𝑖 } ) ;

6

5.1

Experimental Setup

Datasets and queries. We evaluate all methods on seven datasets of native unit-length embeddings, covering cardinalities from 500K to 10M and dimensionalities from 96 to 4096. Table 2 summarizes their characteristics. For Wikipedia-Qwen [58], OpenAI-1536 [36], and OpenAI-3072 [37], we hold out 1K vectors from each original collection as a strictly disjoint query set. MSMARCO-1M and MSMARCO-10M are randomly sampled from the 113.5M-passage collection and use its independent real queries [8]. HotpotQA uses its complete corpus and official queries [7], while Yandex Deep-10M

Query Processing

The layer-specific parameters 𝜅 ℓ affect the query result through the graph topology constructed at each layer, but they are deliberately absent from the query-time scoring function. As shown in Section 4.1, changing 𝜅 ℓ adds the same constant to all squared 9

Table 2: Dataset statistics. Dataset

Data Size

Dim.

Query Size

Data Type

Wikipedia-Qwen OpenAI-1536 OpenAI-3072 MSMARCO-1M HotpotQA Yandex Deep-10M MSMARCO-10M

500,000 1,000,000 1,000,000 1,000,000 5,233,329 10,000,000 10,000,000

4096 1536 3072 1024 1024 96 1024

1,000 1,000 1,000 1,000 1,000 1,000 1,000

Text Text Text Text Text Image Text

we report index construction time to measure indexing efficiency and index memory footprint to quantify storage overhead. The footprint includes all artifacts retained in the deployed index, including transformed vectors, but excludes temporary construction artifacts. Parameter settings. We use the authors’ open-source implementations of ip-NSW [27], ip-NSW+ [23], MAG [59], and PSP [60]; since NAPG has no public implementation, we implement it following the original paper. For all methods, we adopt the recommended parameters from their papers or the default settings of their codebases. Specifically, ip-NSW uses 𝑀 = 32 and efConstruction = 1024; ip-NSW+ additionally builds an auxiliary cosine graph with 𝑀 = 10 and efConstruction = 100; NAPG uses 𝑀 = 16, efConstruction = 100, five norm ranges, and 100 samples per range; MAG uses 𝐿 = 60, 𝑅 = 48, 𝐶 = 300, 𝑅IP = 20, 𝑀 = 64, and threshold 8; and PSP uses 𝐾 = 400, 𝐿 = 800, 𝑅 = 40, angle 60◦ , 𝑀 = 5. Both MAG and PSP require an external 𝑘NN graph, which we construct using FAISS NN-Descent with 𝐾 = 400, 𝐿 = 420, and 12 iterations. TANGO 1 uses 𝑀 = 25, efConstruction = 200, 𝜅 base = 63 at the base layer, and 1 𝜅 nav = 15 at all upper layers across all datasets and both TDVS modes. We vary each method’s search parameter over a sufficiently broad range and compare complete Recall–QPS curves. The number of returned results 𝑘 is set to 50 by default.

uses the first 10M Deep1B vectors and its public queries [3]. Every dataset is evaluated using exactly 1,000 queries. Temporal workloads. Since there is no public dataset with native timestamps, we assign each base vector 𝑥𝑖 a timestamp 𝑡𝑖 ∈ [0, 𝐻 ], with larger values denoting newer objects, and model two representative semantic–temporal relationships. The topic-independent workload captures scenarios where timestamps are unrelated to semantic content, such as general-purpose document repositories and continuously collected logs. We construct this workload by drawing each timestamp independently from Uniform(0, 𝐻 ). The topic-correlated workload captures scenarios where semantically related content occurs during similar periods, such as news articles and conversations with large language models, with topics remaining active over limited time windows. We construct this workload by clustering the unit vectors using spherical 𝑘-means, assigning each cluster 𝑐 an activity center 𝜇𝑐 ∼ Uniform(0, 𝐻 ), and sampling its member timestamps from a normal distribution centered at 𝜇𝑐 and truncated to [0, 𝐻 ]. Following common practice, we parameterize exponential decay using the interpretable half-life ℎ, where 𝜆 = ln 2/ℎ, and use 𝛼 to control the semantic–temporal trade-off in additive TDVS. Unless otherwise stated, we set 𝐻 = 𝜏 = 365 days, ℎ = 90 days, 𝛼 = 0.7, adopt the topic-correlated workload, and evaluate both multiplicative and additive TDVS. Subsequent experiments vary the workload type, ℎ, and 𝛼 to comprehensively evaluate TANGO’s robustness. Benchmark methods. This paper presents two solutions to TDVS: adapting state-of-the-art MIPS indexes through STR (Section 2.3) and natively indexing the Chronos geometry with TANGO (Sections 3 and 4). Accordingly, we compare TANGO with five stateof-the-art graph-based MIPS methods adapted to TDVS via STR [5, 6, 24, 28, 46]. These methods span major advances in graph-based MIPS: ip-NSW [28] directly constructs a non-metric inner-product graph; ip-NSW+ [24] introduces angular navigation; NAPG [46] adopts norm-adjusted graph construction; MAG [5] combines innerproduct and Euclidean geometries; and PSP [6] exploits queryscaled nearest-neighbor geometry with spherical pathways. Together, they cover foundational and recent state-of-the-art designs, forming strong baselines for both multiplicative and additive TDVS. b tdvs (𝑞, 𝜏) Evaluation measures. For each query (𝑞, 𝜏) ∈ Q, let N 𝑘 tdvs and N𝑘 (𝑞, 𝜏) denote the returned and exact top-𝑘. We measure anbtdvs (𝑞,𝜏 )∩N tdvs (𝑞,𝜏 ) N Í 𝑘 swer quality by Recall@𝑘 = | Q1 | (𝑞,𝜏 ) ∈ Q 𝑘 , and 𝑘 query efficiency by queries per second (QPS). We sweep the search budget to obtain complete Recall–QPS curves. For index evaluation,

5.2

Overall Performance

5.2.1 Query Performance. Figures 5 and 7 present the Recall–QPS trade-offs of all methods for multiplicative and additive TDVS. The four representative datasets cover dimensionalities from 96 to 4096, and dataset sizes from 500K to 10M. TANGO consistently achieves the best query performance under both TDVS modes across all datasets. Compared with the best-performing competitor, TANGO achieves speedups of up to 3.06× (recall 0.95) and 3.50× (recall 0.99) under multiplicative TDVS; the corresponding gains under additive TDVS are 2.41× and 2.44×. MAG and PSP fail to reach high recall on Wikipedia-Qwen and HotpotQA under multiplicative TDVS because dispersed true neighbors weaken local connectivity in their semantic Euclidean graphs, limiting navigation and pruning. The results show that TANGO’s advantages are more pronounced on high-dimensional and large-scale datasets, with the performance gap over the baselines often widening in the high-recall region. This suggests that TANGO’s TDVS-native graph geometry is especially beneficial when high dimensionality, large scale, or stringent recall makes graph navigation more challenging. 5.2.2 Index Performance. Figures 6 and 8 report the index construction time and index memory footprint under multiplicative TDVS across all seven datasets; additive TDVS exhibits similar trends, so we present only the multiplicative setting to save space. TANGO is the fastest to construct on every dataset, outperforming the fastest competitor by up to 4.05×. This advantage stems from directly encoding temporal awareness into a compact unified graph, avoiding the expensive candidate expansion, auxiliary graphs, and 𝑘NN-graph-based multi-stage pruning and refinement required by the competitors. All methods have similar memory footprints because they all employ sparse graph structures with comparable connectivity budgets, leaving only limited differences in methodspecific auxiliary storage. Overall, TANGO delivers substantially 10







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5.3.3 Impact of the Semantic–Temporal Weight. Figure 11 varies 𝛼 ∈ {0.3, 0.5, 0.7, 0.9} under additive TDVS. A smaller 𝛼 assigns more weight to temporal freshness, whereas a larger 𝛼 emphasizes semantic similarity. TANGO achieves the best query performance for every value of 𝛼. At recall 0.95, TANGO achieves speedups of 1.48–2.43× over the best-performing competitor. The largest gain occurs at 𝛼 = 0.3 and generally narrows as 𝛼 increases, consistent with the objective becoming closer to static semantic search. Nevertheless, TANGO remains 1.55× faster even at 𝛼 = 0.9. These results demonstrate TANGO’s robustness across different semantic– temporal weights under additive TDVS.

faster index construction while maintaining a comparable memory footprint, demonstrating superior overall index performance.

5.3

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Robustness to Temporal Settings

We next examine TANGO’s robustness to temporal settings by varying the timestamp distribution and half-life ℎ under both TDVS modes, and the semantic–temporal weight 𝛼 under additive TDVS. 5.3.1 Impact of Timestamp Distribution. Figure 9 evaluates the topic-independent setting, complementing the topic-correlated setting used in Figures 5 and 7. In the default topic-correlated workload, vectors within each semantic cluster have timestamps sampled around a shared activity center 𝜇𝑐 . Here, we remove this semantic– temporal correlation by sampling each timestamp independently from Uniform(0, 𝐻 ), regardless of cluster membership (i.e., semantic topics). This weakens the alignment between semantic and temporal neighborhoods and produces a different search geometry. TANGO achieves the best query performance under both TDVS modes. At recall 0.99, it is 2.76× and 2.02× faster than the bestperforming competitor under multiplicative and additive TDVS, respectively. This demonstrates TANGO’s robustness across both topic-independent and topic-correlated temporal workloads.

5.4

Online Insertion Performance

Figure 12 shows TANGO’s online insertion performance on the MSMARCO-10M dataset under both TDVS modes. We build the initial index on the oldest 50% of the vectors and then insert the remaining 50% with a single thread in five batches, each containing the next 10% in timestamp order. As shown in Figure 12(a), insertion throughput remains stable and even improves slightly as the index grows, because later arrivals benefit from increasingly well-covered recent-time neighborhoods, facilitating efficient local graph updates. Figure 12(b) further shows that TANGO maintains 11

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and evaluates the induced distances directly from the original embeddings and timestamps. Unlike prior kernel-based vector search approaches, Chronos requires neither kernel approximation nor explicit high-dimensional transformed representations, thereby avoiding both approximation error and feature-expansion overhead while preserving an exact metric formulation for TDVS-native indexing. Graph-based vector indexes. Vector-search techniques span several major families, including locality-sensitive hashing [48, 52, 54], vector quantization [16, 17, 20], subspace collision [51, 53], and graph [2, 15, 25]. Graph-based indexes are the most widely adopted in practice because they offer strong recall–efficiency trade-offs across diverse workloads [2, 50]. Representative ANNS graph indexes include HNSW [25], NSG [15], and DiskANN [45]. For MIPS graph indexes, ip-NSW [28] directly constructs an inner-product graph, while ip-NSW+ [24], NAPG [46], MAG [5], and PSP [6] improve navigation through angular, norm-aware, hybrid-metric, or query-scaled geometries. These methods are designed for a fixed semantic objective. TANGO introduces a TDVS-native hierarchy that assigns query-equivalent TimeLift geometries to different graph layers, with traversal guided by the exact TDVS score, thereby supporting both temporal locality and long-range semantic navigation.

RELATED WORK

Time-aware retrieval and vector search. Temporal information has been incorporated into information retrieval through documentevolution modeling, recency-aware ranking, and temporal query modeling [9, 13, 26]. Recent work on dynamic question answering, temporal RAG, and agent memory emphasizes retrieving current evidence [21, 34, 39, 56], while EvoWiki and HoH evaluate the effects of evolving or outdated knowledge [33, 47]. For time-aware vector search, existing approaches mainly follow two paradigms. Filter-based methods use temporal predicates, time ranges, or a specified time to define a hard eligible set [14, 18, 35, 49]. Rerankbased methods first retrieve candidates under a static semantic objective and then incorporate recency through score fusion or post-retrieval reranking [9, 32, 39]. In contrast, TDVS keeps all items eligible and incorporates continuous temporal decay directly into the vector-search objective, providing a native search formulation for freshness-sensitive workloads in which relevance evolves continuously over time. Kernel and feature mapping. A kernel is a similarity function that can be interpreted as an inner product after mapping objects into a possibly high-dimensional feature space, allowing complex relationships to be analyzed using standard geometric tools [1]. Since this feature space may be implicit or infinite-dimensional, methods such as random Fourier features approximate certain kernels with explicit finite-dimensional vectors [40]. In vector search, kernelized LSH supports approximate nearest-neighbor search under implicit kernel similarities [22], while MIPS reductions construct explicit query and data mappings that enable existing hashing or nearest-neighbor indexes to preserve inner-product rankings [30, 44]. Chronos introduces a novel implicit kernel-based metricization of the complete TDVS objective. It employs the Laplacian temporal kernel to derive an exact anchor-free TDVS metric

7

CONCLUSIONS

In this paper, we formalized time-decayed vector search (TDVS), which jointly models semantic similarity and temporal freshness. We derived STR as an exact reduction that enables existing MIPS indexes to support TDVS, and further presented Chronos, a TDVSnative metric framework with Query-Orthogonal TimeLift for controllable semantic–temporal geometry. Building on Chronos, we developed TANGO, a hierarchical graph index that combines temporal locality with long-range semantic connectivity and supports efficient online insertion. Extensive experiments on seven real-world datasets demonstrated that TANGO consistently outperforms state-of-the-art graph-based methods in query and index performance and remains robust across diverse temporal settings. 12

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