Enhancing anomaly detection with topology-aware autoencoders

I. INTRODUCTION

• Autoencoders can detect deviations from learned collider backgrounds without predefined signal hypotheses, but latent-space topology constrains their effectiveness.The motivation is anomaly detection for physics beyond the Standard Model, where the new-physics signal may be unknown.Source: §I · p. 1
• Matching the latent manifold to the known topology of phase-space data can minimize reconstruction distortions and improve signal-background discrimination.Energy-momentum conservation gives the background phase space non-trivial manifold structure.Source: §I · p. 1
• The study compares spherical S^n, product S^2 ⊗ S^2, and projective RP^2 latent spaces with standard Euclidean representations on toy and collider datasets.The comparison quantifies effects on reconstruction fidelity, reconstruction error, and anomaly-detection sensitivity.Source: §I · p. 1
• Appropriately chosen latent topologies improve separation between background and anomalous events, especially when the anomaly has a different intrinsic manifold structure.The approach is validated on simulated hadronic top-quark decays, where matching the background structure reduces false positives and spurious reconstruction errors.Source: §I · p. 1
• The paper develops the motivation and mathematical basis for topological priors, constructs spherical and projective latent spaces, and tests them on toy data and simulated top-quark decays.The realistic application compares autoencoders for detecting anomalous three-body decays.Source: §I · p. 2

II. NECESSITY OF TOPOLOGICAL PRIORS

• The analysis treats phase-space manifolds as topological manifolds and focuses on cases that are not homeomorphic to Euclidean R^k.This framing relaxes smoothness to continuity while retaining the global topological structure relevant to representation learning.Source: §II · p. 2
• The primary signal setting assumes the signal manifold is out-of-distribution rather than fully contained within the background manifold.If the signal were a background submanifold, neural-network interpolation could allow efficient signal reconstruction.Source: §II · p. 2

A. Topological Obstructions

• The paper assumes background data lies on an intrinsic k-dimensional manifold and typically uses latent dimension k ≤ M < N, with larger latent dimensions making outlier reconstruction easier.The usual expectation that M = k is optimal fails when the data manifold has non-trivial topology.Source: A. Topological Obstructions · p. 2
• A trivial k-dimensional latent space can produce spurious anomaly regions because one global Euclidean chart cannot faithfully represent a non-trivial background manifold.The resulting high reconstruction errors occur even when encoder and decoder capacity is otherwise sufficient.Source: A. Topological Obstructions · pp. 2–3
• Universal-approximation arguments are applied to compact subsets of R^N identified with the intrinsic data manifold, rather than to all of the ambient space.The stated mathematical qualification arises because standard proofs use an N-dimensional closed unit hypercube as the function domain.Source: A. Topological Obstructions · p. 2
• If the latent manifold is homeomorphic to the background manifold, encoder-decoder maps can represent the background globally and faithfully.This construction removes the topological obstruction in principle and can enhance anomaly detection.Source: A. Topological Obstructions · p. 3

B. Topologies of momentum data

• Fixed-energy and momentum-conserving decays of n distinguishable particles produce phase-space manifolds with spherical topology S^(3n−4), including S^5 for three-body examples.The examples include massive three-particle decay and e+e− → μ+μ−γ at fixed collision energy.Source: B. Topologies of momentum data · p. 3
• Sequential decays generate products of spheres, such as S^2 ⊗ S^2 for leptonic top decay and S^5 ⊗ S^2 ⊗ S^2 for a multistage collider process.Each two-body decay contributes an S^2 factor in the listed examples.Source: B. Topologies of momentum data · pp. 3–4
• Indistinguishable final-state particles identify antipodal momentum configurations, producing an RP^2 factor and yielding S^2 ⊗ RP^2 for hadronic top decay.The projective structure follows from exchanging the two indistinguishable W-decay jets.Source: B. Topologies of momentum data · p. 4

Network-level requirement

• Because dense networks use multiple hidden representations, satisfying the minimum Euclidean embedding dimension alone is insufficient; the latent layer must propagate a global embedding through the decoder.The construction therefore targets the topology of the bottleneck rather than only the width of hidden layers.Source: §III · p. 4

Latent-space design

• The latent manifold is chosen with k ≤ k_l < k′ while still admitting a global embedding of the background data manifold D_k.This separates the background’s embedding requirement from the higher-dimensional signal manifold that should remain difficult to represent locally.Source: §III · p. 4
• A two-layer bottleneck uses a preparation layer followed by a latent layer to realize a non-trivial manifold as an embedding into Euclidean neuron space.The preparation values are y_i, while latent values z_i encode the constrained manifold.Source: §III · p. 4

Explicit manifold constructions

• Spherical and product manifolds are constructed by normalizing neuron groups, including separate normalizations for the factors of S^n1 × S^n2.The product construction repeats the spherical construction across neuron groups, with n_1+n_2+2 neurons in the preparation and latent layers.Source: §III · p. 4
• The projective latent space RP^2 uses its standard embedding into R^4, identifying opposite points so (y_1,y_2,y_3) and −(y_1,y_2,y_3) map identically.The map is differentiable except at y_1=y_2=y_3=0.Source: §III · p. 4

Experimental setup

• The study uses symmetric autoencoders with progressively narrower hidden layers and compares trivial latent spaces with dimensions below and above the manifold’s minimum global embedding dimension.The networks use 1024, 512, 256, 128, and 64 encoder nodes before the latent output, with ReLU hidden activations and tanh at the latent layer.Source: §IV · p. 5

Reconstruction comparison

• R^2-AE cannot fit all points on the sphere, whereas R^3-AE and S^2-AE fit the entire S^2, including the ten validation samples with the largest errors.For R^3-AE and S^2-AE, worst-point inputs and outputs completely overlap in the projected comparison.Source: A. 2-sphere embedded in R3 · p. 5

Global-fit diagnostic

• The loss-versus-distance plot for R2-AE shows high loss near the largest-error point x0 and low loss farther away, unlike the corresponding plots for S2-AE and R2-AE.This behavior is expected because removing a single point from Sn yields a space homeomorphic to Rn, allowing Rn-AE to fit data elsewhere accurately.Source: A. 2-sphere embedded in R3 · p. 5

Anomaly-detection implication

• S^2-AE preserves the background’s intrinsic dimension while clearing the topological obstruction, whereas R^3-AE uses a higher intrinsic latent dimension that may weaken anomaly detection for low-dimensional signals.The realistic collider experiments explicitly test this anomaly-detection implication.Source: A. 2-sphere embedded in R3 · p. 5

Dataset construction

• The product manifold is generated from two independently embedded 2-spheres and placed in R^9, closely mimicking the parton-level sequential two-body top-quark decay background.The resulting components are denoted x_i for i ∈ {1,2,...,9}.Source: B. S2 ⊗S2 embedded in R9 · pp. 5, 7

Reconstruction comparison

• S^2 ⊗ S^2-AE reconstructs the validation data efficiently, with overlapping inputs and outputs even for the ten worst samples, using a network of about 1k parameters.R^3-AE performs worst, R^4-AE is better but retains large errors, and R^5-AE also reconstructs efficiently because S^2 ⊗ S^2 embeds in R^5.Source: B. S2 ⊗S2 embedded in R9 · p. 7

Experimental setup

• The experiment trains R^3-, R^4-, R^5-, and S^2 ⊗ S^2-autoencoders on 400k training and 100k validation samples.The product-topology latent layer has six extrinsic dimensions but four degrees of freedom after independently normalizing its two triplets.Source: B. S2 ⊗S2 embedded in R9 · p. 7

Topological explanation

• R^3-AE fails because S^2 ⊗ S^2 has no well-behaved local chart in R^3, while R^4-AE remains globally inadequate because R^4 cannot embed the product manifold.The comparison distinguishes local-chart sufficiency from the stronger requirement of a global embedding.Source: B. S2 ⊗S2 embedded in R9 · p. 7

• The evaluation targets hadronic top-quark decays, with sequential two-body decays as background and effective four-point three-body decays as possible signal.Source: §V · p. 8

• Background and signal events are generated at 14 TeV with MadGraph5_aMC@NLO, then reconstructed samples receive Pythia8 showering and hadronisation.The signal uses SMEFTsim and forces a non-resonant three-body top decay, while B mesons remain stable for b-jet tagging.Source: A. Event Simulation and Baseline… · p. 8
• Events require missing transverse energy of at least 50 GeV, one negatively charged lepton within |η| ≤3, at least four jets, and at least two b-tagged jets.Jets use anti-kt with R = 0.4 and p_T^min = 40 GeV; candidate tops are retained when a reconstructed mass lies within 15 GeV of 173 GeV.Source: A. Event Simulation and Baseline… · p. 8
• The datasets contain 300k reconstructed background samples and 500k parton-level samples, with three-momenta of top-decay products in the top rest frame embedded in R9.Features are z-score standardised using background means and standard deviations.Source: A. Event Simulation and Baseline… · p. 8

• The idealised background manifold is S2 ⊗ S2, or S2 ⊗ RP2 when the two W-decay jets are treated as indistinguishable, while the signal manifold is S5.The study therefore tests R^k autoencoders for k ∈ {3, 4, 5} and latent topologies S2 ⊗ S2, S2 ⊗ RP2, and S5.Source: B. Network Architecture and Training · p. 8
• Figure 5 compares reconstructed invariant-mass distributions for W and top decay products with their parton-level true distributions.Source: Figure 5 · p. 9
• All autoencoders share a 1.4-million-parameter architecture and are trained on background data with mean squared error, Adam optimisation, learning-rate reduction, and early stopping.Each network is trained ten times from random initialisation for up to 300 epochs with batches of 256.Source: B. Network Architecture and Training · p. 9

• R5-AE and S5-AE achieve at least an order-of-magnitude lower reconstruction error than other topologies, but their AUCs are not better.A five-dimensional bottleneck can locally reconstruct signal data without providing an effective global representation for anomaly detection.Source: C. Results · p. 9
• Reconstructed data produce greater background-signal overlap than parton-level data, reducing the S5-AE AUC relative to the parton-level case.The broader reconstructed W-mass peak weakens anomaly separation after showering and hadronisation.Source: C. Results · p. 9
• All four-dimensional latent topologies have the best validation losses, with S2 ⊗ S2 lowest and S2 ⊗ RP2 second lowest.The result supports the simplified four-dimensional background-manifold assumptions, while RP2 may be harder to optimise because of its more complicated structure.Source: C. Results · p. 9
• Tables I and II report validation-loss statistics and signal AUCs for parton-level and reconstructed datasets across the tested latent topologies.Source: Table I · p. 10, Table II · p. 10
• Four-dimensional non-trivial topologies generally provide better anomaly detection than Euclidean embeddings, confirming the benefit of matching the background manifold topology.The S2 ⊗ RP2 model is nominally better on reconstructed data and identical within precision to S2 ⊗ S2 on parton-level data.Source: C. Results · pp. 9–10

• The topology-aware framework matches latent manifolds to physical background topology, mitigating obstructions that cause spurious reconstruction errors in conventional autoencoders.The study considers manifolds including S^n, S^n ⊗ S^m, and RP^2.Source: §VI · p. 10
• Explicitly using an S^2 ⊗ S^2 latent space achieved near-perfect reconstruction where an R^4 autoencoder struggled on S^2 ⊗ S^2 data.The Euclidean mismatch increased the risk of classifying physically allowed background events as anomalous.Source: §VI · pp. 10–11
• For simulated hadronic top-quark decays, the S^2 ⊗ RP^2 topology achieved AUC 0.7321 ± 0.0037 versus 0.7246 ± 0.0097 for an R^4 autoencoder.An R^5 latent space instead achieved AUC 0.549 ± 0.0015.Source: §VI · p. 11
• Higher-dimensional trivial latent spaces can reconstruct both background and signal efficiently, degrading anomaly detection performance.This result motivates selecting latent topology rather than relying only on unconstrained latent dimensionality.Source: §VI · p. 11
• The results connect data topology with deep-learning representations and provide a framework for incorporating physically relevant priors into anomaly-detection architectures.The stated goal is more robust and interpretable anomaly detection in experimental particle physics.Source: §VI · p. 11
• The current approach assumes that the background topology is known a priori; future work proposes adapting latent topology dynamically when it is not.Extensions to more complex event topologies and multidimensional manifolds are also proposed.Source: §VI · p. 11