VACE: L EARNING G EOMETRICALLY S TRUCTURED R EPRESENTATIONS FOR T IME S ERIES A NOMALY D ETECTION
arXiv:2605.23504v1 [cs.LG] 22 May 2026
Alberto D. Cencillo1 [email protected]
Leonardo Concepción1 [email protected]
Isaac Triguero2,1 [email protected]
Julián Luengo2,1 [email protected] 1
Andalusian Research Institute in Data Science and Computational Intelligence (DaSCI) 2 Department of Computer Science and Artificial Intelligence (DECSAI), University of Granada, Granada, 18071, Spain
A BSTRACT Anomaly detection in multivariate time series is a critical task across a wide range of real-world applications, where abnormal behaviour is rare, labels are unavailable, and the cost of a miss is high. The central challenge is learning a characterisation of normality precise enough to flag deviations. Representation self-supervised learning, typically through contrastive approaches, addresses this by embedding temporal patches into a latent space where normality occupies a well-defined region, with anomalies detected by geometric deviation. However, contrastive approaches shape this space indirectly through pair-sampling heuristics, providing no explicit control over the geometric structure that distance-based scoring requires. This means how tightly normal representations are grouped, and whether distances are directionally meaningful. We present VACE (Velocity-Aligned Channel Embeddings), a self-supervised anomaly detection method that represents normality as a compact, directionally coherent region in the embedding space. To this end, VACE trains a channel-aware encoder through a velocity-consistency objective, with no negatives and no synthetic anomalies, so that normal trajectories are locally smooth and aligned. At test time, a Mahalanobis positional score and a velocity-bank directional score are combined multiplicatively, flagging points that are simultaneously off-distribution and dynamically atypical. Despite its simplicity, VACE achieves state-of-the-art performance on TSB-AD-M under rigorous evaluation, significantly outperforming more complex methods trained on substantially larger budgets.
1
Introduction
The proliferation of multivariate sensor data in industrial, medical, and infrastructure systems has made Multivariate Time Series Anomaly Detection (MTSAD) a critical component of automated monitoring pipelines. In industrial systems, subtle deviations in sensor readings can indicate abnormal operating regimes [1]. In server infrastructures, shifts in resource usage often accompany performance degradation. In medical monitoring, irregular physiological patterns may reflect clinically relevant changes in patient state [2]. Across these domains, normal behaviour is abundant and structured, whereas anomalies are rare, heterogeneous, and costly to miss. This fundamental asymmetry has led to approaches that learn exclusively from normal data, framing anomaly detection as the problem of modelling normality and flagging deviations as anomalies [3]. In MTSAD, normality is not a fixed, well-defined concept: it depends on the operating regime and evolves over time. A localised deviation in a single channel or a sudden shift in global system dynamics can each be anomalous [4], yet whether any of these constitutes a true anomaly is determined by what the system considers normal at that moment. Detecting anomalies, therefore, requires a representation of normality that is stable, bounded, and concrete enough to serve as a reference. However, raw input space cannot provide this property, given its inherent variability.
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
Unstructured Embedding
Task-aligned Embedding
... ...
...
(a) A sliding window extracts N overlapping patches from the multivariate time series X. Each patch xi is encoded by fθ into an embedding vector zi .
(b) A fragmented, directionally incoherent distribution (left) provides no reliable geometric reference for anomaly scoring (red). A compact trajectory (right) makes both positional and directional deviation well-defined signals.
Figure 1: Patch embeddings form a trajectory whose geometry determines anomaly detectability.
Representation learning [5] offers a principled response to this challenge. Originally motivated in computer vision and natural language processing, these methods map raw inputs into an embedding space where the structure implicit in the original data becomes tractable [6]. More recently, they have been applied to MTSAD [7], where a time series is divided into temporal patches that are mapped into an embedding space, as shown in Figure 1a, and normality is characterised by the geometry of the resulting distribution. Anomalies are then scored by a notion of distance from the learned normal region [8] rather than from a reconstruction target. Since anomaly labels are rarely available in practice, this setting falls naturally within the Self-Supervised Learning (SSL) paradigm, where representations are learned from the structure of the data itself [9]. This framing is powerful in principle, but existing methods leave two fundamental questions unanswered [10]. First, how should the encoder be trained so that the embedding space acquires geometric properties like compactness or directional coherence that a distance-based scorer depends on? Second, how should position and motion through the embedding space be jointly scored, given that anomalies can manifest as a point far from the normal distribution, a point moving in an unusual direction, or both? In practice, representation SSL for MTSAD is dominated by contrastive objectives [11]. To answer the first question, these methods reduce the problem to the construction of positive and negative pairs, making the geometry of the embedding space an indirect consequence of sampling heuristics. As a result, pair selection (temporal contrast, synthetic negatives, farthest patches) determines which representations are separated, but provides no explicit control over the compactness or directional coherence that the downstream detection task relies on [12]. This introduces three failure modes. On the one hand, treating all non-overlapping normal windows as negatives disperses semantically similar boundary segments, introducing false negatives [13]. On the other hand, synthetic negatives based on fixed perturbations provide limited boundary coverage and lack difficulty diversity. Finally, without explicit compactness constraints, normal and anomalous embeddings can mix near the decision surface [14]. These distortions corrupt the geometry before the scorer even operates. The answer to the second question has received comparatively little attention. Most methods produce a single scalar distance from the normal region, collapsing the richer structure of the trajectory into a point-wise measure. Anomalies that are positionally plausible but dynamically atypical, or dynamically smooth but spatially displaced, are underscored by any single-signal criterion, as Figure 1b illustrates: in a fragmented distribution (left), anomalous points (red) blend with normal boundary segments and resist distance-based separation. In a compact, directionally coherent trajectory (right), both positional and directional deviation become legible signals. We present VACE (Velocity-Aligned Channel Embeddings), a representation SSL method that addresses both questions from a single geometric commitment: the embedding trajectory of a normal time series should be locally smooth and directionally coherent. This commitment drives the method at two levels. First, we embed each multivariate time series through a channel-aware encoder that preserves per-sensor patterns without diluting them across channels, and train it with an objective that directly targets trajectory coherence. In doing so, the resulting space has the geometric structure a distance-based scorer necessitates, rather than relying on instance separation to induce it as contrastive objectives do. Then, we score anomalies with two complementary signals derived from the same geometry: (i) how far an embedding lies from the learned normal distribution, and (ii) how misaligned its local direction of motion is with normal trajectories. Combining both flags anomalies only when position and direction are simultaneously unusual, a conjunction that neither signal captures alone. 2
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
These design choices are realised in two concrete contributions: 1. A velocity-consistency pretext objective (Lvel ) that directly shapes the embedding space with awareness of the downstream scorer, replacing pair sampling with a geometric prior on trajectory smoothness. A channel-aware encoder based on depthwise-separable convolutions ensures that per-channel anomaly signals are preserved before cross-channel mixing, so that the resulting geometry remains legible at the sensor level. 2. A multiplicative state and dynamics scorer that jointly penalises positional and directional deviation, flagging anomalies only when both signals are elevated. We evaluate VACE on TSB-AD-M [15], a large-scale and comprehensive benchmark for MTSAD, where it achieves state-of-the-art performance. Code is available on an anonymous repository: https://github.com/ari-dasci/ S-VACE.
2
Related Work
The TSAD literature can be broadly organised into three paradigms according to how normality is represented and deviation is measured [16]. Forecasting-based methods define anomalies as deviations between observed values and the predictions of a model trained on normal data. Early approaches used recurrent architectures as one-step predictors, scoring anomalies by prediction error [17, 18]. DeepAnT [19] extended this to convolutional predictors under the same residual-scoring paradigm. More recently, forecasting foundation models like LagLlama [20], TimesFM [21] or Chronos [22] have been adapted to TSAD by treating large prediction residuals as anomaly signals. Faber et al. [23] apply the xLSTM architecture [24] under a multi-step forecasting objective. KAN-AD [25] leverages Kolmogorov–Arnold Networks to model the smooth compositional structure of normal time series. Reconstruction-based methods train a model to compress and reconstruct normal inputs, scoring anomalies by reconstruction error. Autoencoders [26] and variational formulations [27] established this paradigm; GAN-based variants [28] extended it with adversarial training. Transformer architectures introduced association-discrepancy scoring as a proxy for reconstruction fidelity [29], and general-purpose models such as TimesNet [30] and the foundation model MOMENT [31] have since been applied within this framework. Representation-based methods map the time series into an embedding space and characterise normality geometrically, scoring anomalies by distance from the learned normal region. TS2Vec [11] learns general-purpose temporal representations via hierarchical contrastive learning, using non-overlapping segments as implicit negatives. CAE-AD [32] combines a contrastive objective with reconstruction at instance and contextual granularity, though inference still relies on reconstruction error. CARLA [8] treats synthetically injected anomalies as hard negatives and scores by nearest-neighbour distance in the resulting space. PaAno [7] adopts a farthest-patch objective to spread the normal distribution and scores via a memory-bank nearest-neighbour criterion. Several recent works have analysed the structural limits of contrastive training in this setting. Yu et al. [13] show that under normal-only training, treating all non-overlapping windows as negatives disperses semantically similar boundary segments, introducing false negatives that distort the decision boundary. Xie et al. [14] identify complementary failure modes: synthetic negatives from fixed perturbation heuristics lack sufficient boundary coverage and difficulty diversity to shape the decision surface, and the absence of explicit compactness constraints on the normal region allows normal and anomalous embeddings to mix near the boundary.
3
Method
As illustrated in Figure 2, we model a multivariate time series as a trajectory in a learned latent space, where normal behaviour corresponds to smooth and coherent motion. Our approach, VACE, constructs this trajectory by encoding overlapping temporal patches and explicitly shaping their evolution through a velocity-consistency objective. This results in latent representations that are not only spatially structured but also temporally regular, enabling the detection of anomalies as deviations in both position and direction of motion. 3.1
Geometric Motivation
Let X ∈ RT ×D denote a multivariate time series of length T with D channels, and let Xa:b ∈ R(b−a)×D denote the contiguous segment spanning timesteps a, . . . , b − 1. Fixing a patch length P , we extract N = T − P + 1 overlapping patches with unit stride, 3
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
Training
Time Training Patches
Trained Encoder
... ...
...
Inference Test Patches
Positional Score
x
Directional Score
Anomaly Score
...
Figure 2: VACE architecture: At training, overlapping patches are encoded by fθ under the velocity-consistency objective Lvel , which shapes the normal embedding trajectory Z into a compact, directionally coherent structure. At inference, test patches are embedded and scored by their positional deviation from Z and the angular misalignment of their local velocity with the normal velocity bank.
xi = Xi: i+P ∈ RP ×D ,
i = 1, . . . , N,
(1)
and encode each patch into a vector zi = fθ (xi ) ∈ Rdz ,
(2)
where fθ : RP ×D → Rdz is a parametric encoder with learnable parameters θ and dz is the embedding dimension, and the ordered tuple Z = (z1 , . . . , zN ) is the normal embedding trajectory of X, as illustrated in Figure 1a, Continuity by construction. Using unit stride produces maximal overlap between consecutive patches: xi and xi+1 share P − 1 timesteps and differ only at their boundary elements. As a result, when the underlying time series evolves smoothly, consecutive patches are strongly correlated in input space. Since the encoder fθ is continuous, this local similarity is preserved in the latent space, leading to embeddings zi = fθ (xi ) that vary smoothly across time. In practice, this induces the trajectory Z that evolves gradually, without abrupt transitions for normal data. Our goal is to exploit this structure by encouraging latent trajectories that remain coherent and evolve smoothly over time, so that deviations from this behaviour can be effectively detected. Decomposition of the anomaly score. Given a test patch embedding z⋆ , we define a positional score sp (z⋆ ), measuring distance from z⋆ to the training distribution under a covariance-adapted metric, and a directional score sd (z⋆ ), measuring how misaligned its local velocity is with the bank of normal training velocities. The final score is their multiplicative combination. The encoder is trained with Lvel to produce the geometry exploited by both scores. 3.2
Channel-Aware Patch Encoder
When employing a 1D-CNN for patch encoding, there are two natural variants. A shared-kernel encoder processes all channels jointly from the first layer: cross-channel relationships are preserved, but a single deviating channel can be 4
Features
1D-CNN
Time
1D-CNN
Features
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
1D-CNN
Features
Features
Time 1D-CNN
Time
(a) Channel-aware patch encoder.
(b) Velocity-consistency loss.
Figure 3: Detailed view of two core components: (a) the channel-aware patch encoder and (b) the velocity-consistency loss. averaged against the majority of normal ones, potentially masking channel-local anomalies in the embedding. A channelindependent encoder applies separate filters per channel [33]: per-channel signals are preserved, but inter-channel structure is discarded, making relational anomalies invisible. For the embedding trajectory Z to be geometrically meaningful, both types of event must remain legible in Rdz , a requirement neither variant alone satisfies. We therefore adopt a two-stage design that separates the two operations: per-channel convolutions first extract local features without cross-channel interference, and a pointwise layer then integrates them into a joint representation. Delayed cross-channel mixing. As depicted in Figure 3a, we adopt the “extract per-channel features first, then mix” principle also leveraged by ModernTCN [34], and implement it with depthwise-separable convolutions [35]. The construction provides two properties simultaneously: (i) a depthwise first stage in which each input channel is processed by its own kernel bank, so per-channel features are captured before any combination takes place; and (ii) a pointwise (1×1) convolution that mixes these enlarged per-channel features into a joint representation. Cross-channel mixing is therefore delayed until per-channel information has already been elevated to a richer feature space. 3.3
Velocity-Guided Representation Learning
We train the encoder fθ using a self-supervised objective designed to enforce a specific geometric property: the embedding trajectory Z induced by normal data should be locally coherent, i.e., exhibit consistent direction of motion over time. Our formulation is inspired by RECURVE [36], which regularises second-order structure (curvature) for boundary detection. Here, we focus on its first-order counterpart, encouraging local rectilinearity of the trajectory. Velocity-consistency loss. For a temporal offset δ, we define the backward and forward normalised velocities at position t as zt − zt−δ zt+δ − zt vtb = , vtf = . (3) max (ϵ, ∥zt − zt−δ ∥2 ) max (ϵ, ∥zt+δ − zt ∥2 ) We minimise their cosine misalignment: Lvel =
N 1 X 1 − ⟨vtb , vtf ⟩ . N t=1
(4)
This loss enforces that, locally, the trajectory follows a consistent direction (see Figure 3b): the vector entering zt is aligned with the one leaving it. Optimum Lvel = 0 is achieved when the trajectory is piecewise linear, since each term (1 − ⟨vtb , vtf ⟩) is non-negative and vanishes if and only if vtb = vtf . Geometry metrics. To characterise the embedding geometry, we report four diagnostics. The participation-ratio effective rank and entropy effective rank [37] are P 2 λi ) ρPR = Pi 2 i λi (
! ρH = exp −
X i
pi log pi
,
λi pi = P j λj
(5)
where λ1 ≥ · · · ≥ λdz are the eigenvalues of the sample covariance Σ. Higher values indicate a more isotropic distribution. We also report the active dimension fraction, i.e. the share of components with explained variance above 5
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
P 1/dz , and the top-1 variance fraction, defined as λ1 / i λi . These metrics are commonly used to assess representation quality [38] and here serve to verify the geometric regime assumed by the scorer and detect collapse [39]. 3.4
Anomaly Scoring via Embedding Geometry
Having trained fθ to produce a trajectory Z that is (i) compact and (ii) locally rectilinear on normal data, the scorer reads off the two geometric signatures left by training: how far a test embedding deviates from the normal region, and how misaligned its local direction of motion is with the bank of normal velocities. Positional score. We fit a Gaussian distribution N (µ, Σ) to the set Z of all embedded training patches. The positional score of a test embedding z⋆ is defined as the squared Mahalanobis distance [40]: sp (z⋆ ) = (z⋆ − µ)⊤ (Σ + εI)−1 (z⋆ − µ),
(6)
−6
with ε = 10 for numerical stability. Mahalanobis reduces to Euclidean distance when Σ ∝ I and upweights deviations along low-variance directions otherwise. The regulariser ε stabilises the inverse when Σ is near-singular. This is a common regime for deep embeddings, where a few directions carry most of the variance; ε prevents a single small eigenvalue from dominating sp . Directional score. To characterise the local dynamics of the embedding trajectory, we employ the velocity vector vtf with temporal offset δ defined in Equation 3. We set δ = P/2, so that the velocity captures dynamics over half a patch horizon. Evaluating this over the training sequence yields a set of velocity vectors {vtf }. We summarise their distribution by computing K prototypes {ci }K i=1 via MiniBatch k-means on the unit sphere [41]. Given a test embedding z⋆ , we compute its associated velocity v⋆f in the same way. We then identify the k nearest prototypes (knp ) to v⋆ in cosine distance. Let Iknp (v⋆f ) ⊂ {1, . . . , K} denote the indices of these prototypes. The directional score is defined as: X 1 sd (z⋆ ) = 1 − ⟨v⋆f , ci ⟩ . (7) knp f i∈Iknp (v⋆ )
By construction, sd measures the deviation of v⋆f from the directions observed during training: normal trajectories produce aligned velocities, while anomalous dynamics lead to larger angular deviations. Composition. Let s̃p and s̃d denote the normalised versions of sp and sd , obtained by rescaling to zero mean and unit variance over the test set. The final anomaly score is defined as: s(z⋆ ) = s̃p (z⋆ ) · 1 + w s̃d (z⋆ ) , w = 1. (8)
4
Experiments
4.1
Experimental Setup
Benchmark Selection. We evaluate on the TSB-AD-M benchmark [15], a curated collection of 200 multivariate time series designed to address known limitations of prior MTSAD evaluation practices, including label leakage and metric sensitivity to the point-adjustment (PA) heuristic [42]. The benchmark covers 17 categories spanning NASA telemetry [1] (MSL, 16 series; SMAP, 27 series), server monitoring [27] (SMD, 22 series), secure water treatment [43] (SWaT, 2 series), and further domains including medical waveforms, industrial processes, and environmental monitoring. Each series is split into temporally adjacent train and test sets; the training set is anomaly-free, and anomaly labels are only provided for the test set. The 200 time series benchmark is divided into an “Eval” set (180 series) and a “Tuning” set (20 series) for hyperparameter optimisation. Evaluation Metrics. We report both threshold-dependent and threshold-free metrics. For threshold-dependent evaluation, we use Point-F1 and Range-F1 [44], the latter weighting full anomaly segments over individual timestamps to better reflect temporal recall. Because thresholding is orthogonal to scoring quality and can mask differences in anomaly ranking [3], we give primary emphasis to threshold-free metrics: AUC-ROC, AUC-PR, and the Volume Under the Surface variants VUS-ROC and VUS-PR [45]. These VUS metrics integrate performance over a range of detection lags and are specifically designed to be robust to the PA inflation problem that affected earlier benchmarks. Compared Models and Resolution Mismatch. Methods that score at the window level, including CARLA [8] and CAAE [14], cannot be fairly evaluated under VUS-PR, which explicitly penalises poor temporal localisation. We 6
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
Table 1: Average results over 10 seeds on TSB-AD-M. Best result in bold, second best underlined. Category Method VUS-PR VUS-ROC AUC-ROC AUC-PR F1 Range-F1 TimesNet DeepAnT OmniAnomaly USAD KAN-AD DADA xLSTMAD-F (MSE) xLSTMAD-R (MSE)
0.19 0.31 0.31 0.30 0.41 0.31 0.35 0.37
0.64 0.76 0.69 0.68 0.75 0.73 0.77 0.72
0.56 0.73 0.65 0.64 0.73 0.69 0.74 0.68
0.13 0.32 0.27 0.26 0.38 0.31 0.35 0.32
0.20 0.37 0.32 0.31 0.42 0.35 0.40 0.38
0.17 0.37 0.37 0.37 0.41 0.25 0.42 0.36
Transformers
CrossAD CATCH AnomalyTransformer DCDetector
0.33 0.30 0.12 0.09
0.77 0.73 0.57 0.56
0.74 0.67 0.52 0.50
0.34 0.24 0.07 0.05
0.38 0.30 0.12 0.10
0.37 0.27 0.14 0.10
Traditional
PCA IForest
0.31 0.20
0.74 0.69
0.70 0.66
0.31 0.19
0.37 0.26
0.29 0.24
Self-Supervised
PaAno VACE
0.43 0.48
0.79 0.78
0.76 0.76
0.38 0.43
0.43 0.46
0.41 0.46
Neural Networks
restrict comparisons to models that produce point-aligned scores. We evaluate against 14 baselines spanning all major paradigms. First, neural network-based methods include TimesNet [30], DeepAnT [19], OmniAnomaly [27], USAD [26], KAN-AD [25], DADA [46], and xLSTMAD [23]. Second, transformer-based approaches comprise CrossAD [47], AnomalyTransformer [29], DCDetector [48] and CATCH [49]. Third, we consider traditional methods such as PCA [50] and IForest [51]. Finally, we include PaAno [7] as our primary self-supervised counterpart and the current state-of-the-art on TSB-AD-M. For all baselines, we report results from the standardised TSB-AD-M evaluation protocol [15]. Implementation Details. The architecture utilises a channel-aware 1D-CNN encoder fθ as detailed in Section 3.2, employing a four-layer structure with varying kernel sizes and a channel expansion factor of ce = 8. We normalise input patches of length P = 96 using RevIN [52], extracting them with unit stride to generate overlapping embeddings. The encoder is trained for 20 iterations under the velocity-consistency objective Lvel with a linearly decaying weight λt , using AdamW with cosine annealing and a batch size of 512. For final scoring, we compute a multiplicative combination (w = 1) of the Mahalanobis distance and a velocity bank, the latter of which is built from up to 500 centroids (10% of training patches) and evaluated using the three nearest prototypes. Point-level scores are obtained by averaging patch scores over all patches covering each timestep. All experiments were conducted on a single NVIDIA RTX 2080 Ti. Hyperparameter sensitivity is analysed in Appendix A.
4.2
Comparative Study
Main results. We evaluate VACE on all 180 time series in the TSB-AD-M evaluation set against 14 competitive baselines, as reported in Table 1. Our method achieves the highest VUS-PR, AUC-PR, F1, and Range-F1; while VUS-ROC and AUC-ROC are comparable to PaAno, the gains in precision–recall metrics indicate more accurate point-wise detection and more temporally coherent identification of anomalous events. Figure 4a confirms this advantage is broadly distributed: VACE outperforms PaAno on 115 of 180 individual series and xLSTMAD on 115 of 180, beating both simultaneously on 79 series against only 29 where both outperform it. Figure 4b reveals a systematic property of the task: detection difficulty increases monotonically as anomaly density decreases, with all methods losing more than half their VUS-PR from the densest to the sparsest bin. Nevertheless, VACE maintains a consistent margin throughout, suggesting the geometric scoring advantage is not dependent on anomaly density. These results substantiate the effectiveness of coupling training and scoring through a shared geometric objective: shaping the embedding trajectory via velocity consistency leads to a representation space in which positional and directional deviations act as complementary, non-redundant anomaly signals. Additional experiments are reported in Appendix B. Notably, VACE reaches this performance with 20 pretext iterations on a single GPU, suggesting that a well-posed geometric objective is a more effective inductive bias than architectural complexity. 7
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
1.0
xLSTMAD PaAno VACE
VUS-PR (mean ± std)
0.8 0.6 0.4 0.2 0.0
0.002 0.009 (n=36)
0.010 0.023 (n=36)
0.025 0.049 (n=36)
0.049 0.088 (n=36)
Anomaly ratio bin (equal-size, 36 series each)
0.090 0.316 (n=36)
(a) Per-series VUS-PR comparison against VACE. (b) Mean VUS-PR across equal-size anomaly-density bins. All methods improve Most points lie below the diagonal. When VACE monotonically as anomaly ratio increases, confirming sparsity as the primary loses to xLSTMAD, the gaps are often larger, indi- difficulty driver. VACE maintains a consistent margin across the full spectrum. cating an advantage of reconstruction-based methods on some sequence-level anomaly series.
Figure 4: Per-series and per-density breakdown of VACE against two baselines. Table 2: Ablation study on the 180-series held-out evaluation set of TSB-AD-M (mean over 10 seeds). ✓/✗ indicate whether each component is active. ∆ is the drop in VUS-PR relative to the full model. Component Metric
Channel Encoder
Velocity Pretext
Mahal. Scoring
Velocity Scoring
VUS-PR
AUC-PR
F1
∆VUS-PR
w/o Channel Encoder w/o Velocity Pretext w/o Mahal. Scoring w/o Velocity Scoring
✗ ✓ ✓ ✓
✓ ✗ ✓ ✓
✓ ✓ ✗∗ ✓
✓ ✓ ✓ ✗
0.438 0.441 0.470 0.472
0.390 0.395 0.421 0.423
0.439 0.438 0.458 0.459
−0.039 −0.037 −0.008 −0.006
Full model
✓
✓
✓
✓
0.478
0.429
0.465
—
Variant
∗
Mahalanobis distance replaced by a memory-bank scorer.
4.3
Model Analysis
Ablation study. Table 2 reports the contribution of each component, measured as VUS-PR drop relative to the full model averaged over ten random seeds on the 180-series held-out evaluation set. Removing the channel-aware encoder produces the single largest drop of 3.9%: without depthwise-separable convolutions, per-channel anomaly signals are diluted by early cross-channel mixing before any discriminative feature can be formed. Removing the velocity pretext yields a drop of 3.7%, validating the central design principle that shaping the trajectory geometry during training translates directly into scoring quality. Replacing Mahalanobis scoring with the memory-bank approach used in PaAno [7] results in a drop of 0.8%: a single Gaussian fit is sufficient precisely because the pretext produces a compact, well-structured distribution, making a non-parametric memory bank unnecessary. Removing velocity scoring leads to a 0.6% drop in performance. This small individual effect, compared to the pretext gap, suggests that the directional signal is partially redundant with the positional one when the trajectory is already well-structured. However, it still provides a consistent and independent source of evidence that the multiplicative combination can exploit. Embedding geometry analysis. Figure 5 characterises the embedding distribution under three configurations across 180 series and 10 seeds, using the geometry metrics introduced in Section 3.3. The full model occupies a structured anisotropic regime, with entropy effective rank 0.115/d, 99.6% of dimensions active, and 55% of variance captured by the leading component. This is precisely the geometry required by the scorer: clear high-variance directions for Mahalanobis distance and dominant motion axes for the velocity bank. Removing the velocity pretext pushes the distribution towards uniformity (ρH /d = 0.143), reducing Mahalanobis to a noisier approximation of isotropic distance. 8
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
0.40
uniform spectrum
Embedding geometry regime (per-series mean, 180 eval series) Mean VUS-PR Full model VUS-PR: 0.478 w/o vel. pretext VUS-PR: 0.441 w/o BatchNormalisation VUS-PR: 0.407
Entropy effective rank ( H/d)
0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
dimensional collapse
0.2
0.4
0.6
Top-1 variance fraction
0.8
1.0
Figure 5: Geometric properties of the embedding distribution (Section 3.3) for each ablation configuration, averaged over 10 seeds and 180 eval series. The velocity pretext places the full model at moderate anisotropy, the regime in which both scorers are most effective. Without the pretext the distribution is too uniform; without BatchNormalisation it collapses onto a single direction. The mean VUS-PR per configuration confirms that the geometry ordering matches the detection ordering.
To contrast this with the opposite failure mode, we also evaluate a variant without BatchNormalisation, which is not included as a performance ablation but serves as a geometric diagnostic. The spectrum collapses to ρH /d = 0.037, the covariance becomes near-singular, and sp loses sensitivity in all but one direction. The 3.7% VUS-PR drop from removing the pretext therefore reflects not only lower performance, but the loss of the geometric structure both scorers depend on. Full numerical diagnostics and a per-category breakdown are analysed in Appendix C.
5
Conclusion
We present VACE, a self-supervised anomaly detection method built on a single geometric commitment: the embedding trajectory of a normal time series should be locally smooth and directionally coherent. Its encoder is trained with an objective directly aligned with the downstream scorer, producing the compactness and directional structure that positional and velocity scoring rely on. Despite its simplicity, VACE achieves state-of-the-art performance on TSB-ADM, outperforming methods that are substantially more complex and trained on larger computational budgets, pointing to geometric structure, rather than scale, as the more effective inductive bias.
6
Acknowledgments
This publication is part of the Project “Ethical, Responsible and General Purpose Artificial Intelligence: Applications In Risk Scenarios” (IAFER) Exp.:TSI-100927-2023-1 funded through the Creation of university-industry research programs (ENIA Programs), aimed at the research and development of artificial intelligence, for its dissemination and education within the framework of the Recovery, Transformation and Resilience Plan from the European Union Next Generation EU through the Ministry for Digital Transformation and the Civil Service.
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VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
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VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
A
Sensitivity Analysis
We assess the sensitivity of VACE to its five main hyperparameters: patch size P , velocity offset δ, number of pretext iterations, velocity weight w, and channel expansion factor ce . All sweeps are conducted on the 20-series tuning subset of TSB-AD-M (disjoint from the 180-series held-out evaluation set used in the main results). Results are stable across a wide range of values; the main sensitivity is to P (small patches hurt substantially) and to the pretext iteration count (both too few and too many degrade performance, with a clear optimum at 20 steps). As depicted in Table 3, default values are highlighted. Table 3: Hyperparameter sensitivity (tuning set, 20 series). Each cell shows VUS-PR value. The default configuration is highlighted. Hyperparameter Patch size P Velocity offset δ Iterations Velocity weight w Channel expansion ce
Low
Default
High
48 (0.464) 24 (0.568) 10 (0.515) 0.5 (0.574) 4 (0.568)
96 (0.577) 48 (0.573) 20 (0.574) 1.0 (0.574) 8 (0.573)
128 (0.578) 96 (0.553) 40 (0.542) 2.0 (0.573) 16 (0.542)
Patch size. Performance degrades sharply for P < 96: small patches carry insufficient temporal context for the velocity-consistency objective to shape a coherent trajectory. Larger patches (P = 128) perform comparably on the tuning set but reduce the number of patches available for fitting the Mahalanobis distribution. Velocity offset. δ = 48 (half the patch horizon) and δ = 24 (quarter-horizon) yield similar VUS-PR. δ = 96 (full patch horizon) degrades slightly, as the velocity vectors span the full patch and become less sensitive to local dynamics. Pretext iterations. Ten pretext steps are insufficient to shape the trajectory geometry, while 30 or more overfit the velocity structure of the tuning series, reducing generalisation. Twenty steps sits at the optimum and is consistent with the finding in the main text that the velocity pretext converges faster than conventional contrastive objectives. Velocity weight. VUS-PR is nearly flat across w ∈ {0.5, 1.0, 2.0}, varying by less than 0.003. The directional score contributes a small but consistent improvement over the positional score alone (confirmed by the ablation in Table 2), and its magnitude is not sensitive to the exact weighting. Channel expansion. ce = 8 and ce = 4 perform similarly; ce = 16 drops notably, consistent with overparameterisation at the depthwise stage.
B
Additional Experiments
Table 4 reports per-category VUS-PR for VACE and two competitive baselines. Overall pattern. VACE leads on 7 of 17 categories. The two largest categories in the evaluation set, SMAP and SVDB, together account for 53 of 180 series, and VACE leads on both; this concentration in large-n categories drives the overall benchmark gap over PaAno despite trailing on several smaller ones. Channel-local fault datasets. Two of the categories where VACE improves most over PaAno are MSL and SMD, both of which contain multivariate sensor recordings where faults commonly affect individual channels. This is consistent with the channel-encoder ablation (Table 2), which shows a drop of 3.9% in VUS-PR when the depthwise-separable encoder is replaced with a standard shared-kernel CNN. The ablation provides direct experimental support for the conclusion that per-channel feature preservation is especially valuable in these two categories; we are cautious about generalising this claim to others where we do not have the same controlled evidence. Hard categories. GHL is the hardest category across all evaluated methods: no model in Table 4 exceeds 0.083 VUS-PR across its 23 series, and VACE scores the lowest. Given that every method struggles equally, we refrain from attributing the difficulty to any specific property of the anomaly type. 13
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
n VACE PaAno xLSTM-AD-R (MSE)
Ov era ll
GH L
CA TS v2
rd dit Ca Cre
OP PO RT UN
PS M
et phn Da
nes is Ge
SW aT
L MS
O
B MI TD
GE CC
SM D
SM AP
B SV D
LT DB
TA O
Ex ath lon
ITY
Table 4: Per-category VUS-PR comparison. Best result per column in bold. VACE is evaluated on the 180-series held-out set (20 tuning series excluded); baseline results are from the TSB-AD-M benchmark [15].
25
11
4
28
25
20
1
11
14
2
1
1
1
7
1
5
23
180
0.928 0.814 0.944
0.692 0.727 0.737
0.621 0.622 0.421
0.612 0.561 0.343
0.586 0.522 0.275
0.434 0.358 0.402
0.416 0.187 0.192
0.396 0.376 0.141
0.344 0.245 0.332
0.273 0.246 0.521
0.271 0.333 0.668
0.261 0.274 0.283
0.220 0.202 0.401
0.108 0.161 0.314
0.057 0.027 0.068
0.051 0.074 0.235
0.010 0.083 0.019
0.478 0.430 0.392
Comparison with xLSTM-AD-R. xLSTM-AD leads on Genesis, SWaT, OPPORTUNITY, and PSM. These are categories where reconstruction-based methods appear to provide a stronger signal; whether this advantage stems from the autoregressive structure, the longer effective context window, or the specific anomaly types present in these datasets is not determined by our experiments. Comparison with TSPulse. TSPulse [53] is a state-of-the-art time series pretrained model, representing a larger resource regime than VACE. We evaluate it independently using the official implementation from the TSB-AD repository [15], under the same protocol applied to all other methods. Due to memory overflow, the OPPORTUNITY category (7 series) could not be evaluated; results are therefore reported over 173 of the 180 evaluation series. Table 5 reports both the fine-tuned (FT) and zero-shot (ZS) variants. The scores obtained under this protocol are higher than those reported in the original TSPulse paper, likely reflecting differences in benchmark composition and evaluation setup; we report our reproduced numbers for consistency with the TSB-AD-M evaluation framework used throughout this work. Under this protocol, VACE outperforms both TSPulse variants on VUS-PR. Table 5: Results on 173 TSB-AD-M evaluation series (OPPORTUNITY excluded). Reproduced from the official TSB-AD implementation. Variant VUS-PR VUS-ROC AUC-ROC AUC-PR F1 Range-F1
C
TSPulse (ZS) TSPulse (FT)
0.40 0.44
0.72 0.76
0.67 0.72
0.35 0.38
0.40 0.43
0.42 0.42
VACE
0.49
0.80
0.77
0.44
0.48
0.46
Geometry Analysis
This section reports the embedding geometry diagnostics in full. Table 6 provides the aggregate metrics for the three configurations (mean ± std over 10 seeds, 180 eval series), complementing the scatter in Figure 5 with precise numerical values. Figure 6 then extends the analysis to the category level, comparing FM, NVP, and NBN across the 17 dataset families in TSB-AD-M, and showing how consistently the geometric separation observed in aggregate arises at the individual category level. Velocity pretext. The pretext task reduces effective rank in 120 of 180 series (67%) and increases nactive in 152 of 180 (84%), from 61.0 to 63.7 active dimensions. These effects are consistent. The objective sharpens dominant principal components and lifts near-zero dimensions above the activity threshold. As a result, the eigenspectrum shifts from near-uniform to a spike-and-tail structure. These results confirm that the resulting geometry is precisely the regime the scorer requires: clear high-variance directions for the Mahalanobis distance to exploit, and a well-populated embedding space with no wasted capacity. Twenty gradient steps of velocity consistency produce this structure reliably across 16 of 17 datasets. Channel encoder. The channel encoder’s effect on geometry is heterogeneous and scales with the number of input channels. For series with two to five channels, the channel encoder leaves the effective rank essentially unchanged (∆ρPR /d = +0.003, near zero), and slightly reduces the top-1 variance fraction (−0.026). This suggests that with very few channels, the depthwise stage has limited per-channel structure to extract before mixing, and the encoder converges to a solution similar to shared-kernel processing. For series with more than 20 channels, the channel encoder consistently reduces effective rank and concentrates variance. 14
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
Figure 6: Per-category geometry diagnostics for the full model, without the velocity pretext, and without BatchNormalisation. The three configurations maintain distinct geometric regimes consistently across all 17 dataset families. Table 6: Embedding geometry diagnostics (mean ± std over 10 seeds, 180-series held-out set).
Eff. rank (ρPR /d)
Eff. rank (ρH /d)
Active dims
Top-1 var. frac.
w/o Velocity Pretext w/o BatchNormalisation
0.081 ± 0.004 0.025 ± 0.001
0.143 ± 0.005 0.037 ± 0.002
0.953 ± 0.006 0.853 ± 0.021
0.493 ± 0.015 0.816 ± 0.014
Full model
0.061 ± 0.002
0.115 ± 0.002
0.996 ± 0.001
0.551 ± 0.010
Variant
The effect is strongest for the highest-channel Table 7: Per-series geometry and detection performance on MSL (14 group (71–248 channels, n = 7, ∆top-1 eval series, mean over 10 seeds). Sorted by ∆VUS-PR. nactive increases = +0.124), where early cross-channel mix- in all 14 series. VUS-PR improves in 8 of 14. ing in the shared-kernel baseline dilutes persensor patterns across many channels before VUS-PR nactive / 64 any discriminative feature can form. The Series NCA FM ∆ NCA FM MSL category, as shown in Table 7, illustrates this division: MSL (55 channels) shows MSL id_1 0.087 0.665 +0.578 47 64 a large increase in active dimensions under 46 64 MSL id_13 0.485 0.792 +0.308 the channel encoder (nactive increases from MSL id_12 0.482 0.713 +0.231 47 64 48.9 to 64.0) with respect to the no channel48 64 MSL id_14 0.567 0.766 +0.198 aware encoder ablation (NCA), indicating MSL id_15 0.490 0.530 +0.041 45 64 that the shared-kernel baseline leaves ≈15 MSL id_4 0.262 0.294 +0.032 52 64 embedding dimensions inactive. MSL id_11 0.095 0.122 +0.027 48 64 MSL id_2 0.107 0.117 +0.010 52 64 BatchNormalisation collapse. Removing MSL id_6 0.071 0.071 0.000 52 64 BatchNormalisation produces a qualitatively MSL id_9 0.068 0.043 −0.025 48 64 different failure than removing the pretext. MSL id_7 0.441 0.384 −0.057 47 64 Effective rank falls to 0.025 across all 17 catMSL id_5 0.142 0.075 −0.067 51 64 egories (0.017–0.046), and nactive drops to MSL id_16 0.302 0.234 −0.068 48 64 54.6 out of 64: roughly nine dimensions beMSL id_8 0.171 0.012 −0.159 51 64 come inactive. The no-pretext variant also loses geometric structure but retains full dimensional occupancy (nactive = 61.0). Collapse compounds two failures. On the one hand, the spectrum becomes concentrated but uninformative. On the other hand, the covariance matrix becomes nearly singular, and its inverse is dominated by near zero eigenvalues rather than meaningful directions. 15
VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection
D
Limitations
Three structural limitations bound the applicability of VACE. First, very long training series produce a large number of overlapping patches, increasing the volume of the embedding point cloud and making the single-Gaussian Mahalanobis fit a noisier approximation of the normal region. Second, the channel-aware encoder processes channels independently before mixing, which is beneficial when anomalies manifest in individual sensors but may discard discriminative inter-channel correlation structure when anomalies are defined by unusual relationships between jointly behaving channels. Third, reconstruction-based methods outperform VACE on a subset of categories where the geometric scoring signal is insufficient; the specific properties of these datasets that favour a predictive characterisation of normality over a geometric one are not determined by our experiments.
E
Broader Impacts
VACE is an anomaly detection method for multivariate time series, with potential positive applications in industrial monitoring, server infrastructure management, and medical signal analysis, where early detection of abnormal behaviour can prevent failures or inform clinical decisions. The method operates on anomaly-free training data and produces anomaly scores without any decision-making authority; deployment decisions remain under human control. We identify no direct path to negative societal applications.
16