One-Class SVM: Smooth CVaR Optimization

• The paper introduces a novel OC-SVM framework that integrates smooth CVaR surrogates with signature-based embeddings to achieve tractable risk calibration.
• The methodology uses shuffle-product identities to derive closed-form polynomial surrogates, leading to explicit error bounds and enhanced hypothesis testing.
• Empirical evaluations in anomalous diffusion and RNA modification detection demonstrate improved type I/II error control and increased detection power over traditional methods.

One-class Support Vector Machine (OC-SVM) algorithms optimising smooth Conditional Value-at-Risk (CVaR) objectives constitute a significant advance in novelty detection within path spaces, connecting sequential data analysis, statistical learning, and probability in function spaces. This class of algorithms exploits signature-based feature embeddings and the shuffle-product structure, enabling closed-form polynomial surrogates for risk-sensitive test statistics and new theoretical guarantees for error control and statistical power in hypothesis testing settings (Gasteratos et al., 2 Dec 2025).

1. Signature-based Features and Smooth CVaR Surrogates

Let $X \sim \mu$ denote a path whose signature $S_N(X)$ is taken up to truncation level $N$. To approximate the positive part $[u]^+$ on $[-K,K]$, a polynomial $Q_n(u) = \sum_{i=0}^n a_i u^i$ is introduced. The smooth CVaR surrogate is then defined by

$$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$

Employing the shuffle-product identity, $(\langle w, S \rangle)^i = \langle w^{\shuffle i}, S \rangle$, Theorem 3.1 shows the surrogate may be rewritten as

$E_\mu\left[ Q_n(\langle w, S(X) \rangle - \rho) \right] = \langle Q_n^{\shuffle}(w - \rho 1), E_\mu[S(X)] \rangle$

where $Q_n^{\shuffle}(\ell) = \sum_{i=0}^n a_i \ell^{\shuffle i} \in (T^{nN}(\mathbb{R}^d))^*$. The result is an explicit polynomial $S_N(X)$0 whose coefficients $S_N(X)$1 depend only on $S_N(X)$2 and shuffle-powers of $S_N(X)$3. This surrogate admits closed-form computation, substantially improving tractability for high-dimensional path data.

2. OC-SVM Formulations and Optimisation Problems

The OC-SVM framework is captured as minimising a regularised CVaR of negative scoring functionals. Given a feature map $S_N(X)$4, typically the truncated signature $S_N(X)$5 or its infinite-level version, the population-level objective reads

$S_N(X)$6

Replacing $S_N(X)$7 by the smooth surrogate yields the smooth-CVaR OC-SVM problem,

$S_N(X)$8

For empirical OC-SVM with finite samples $S_N(X)$9, the unconstrained primal is

$N$0

A constrained quadratic program variant introduces slack variables $N$1: $N$2

3. Dual Formulation and Signature Kernels

The dual form of the constrained OC-SVM is

$N$3

for kernel matrix $N$4. With $N$5, the signature kernel is

$N$6

Given solution $N$7, the primal vector is $N$8 and the test-score for a new path $N$9 is $[u]^+$0. In the smooth-CVaR population version, $[u]^+$1 enters via the closed-form surrogate, replacing empirical averages.

4. Theoretical Error Bounds: Type I and Power

Denoting $[u]^+$2, the following bounds are established:

• Type I Error (Theorem 3.4): If $[u]^+$3 obeys an $[u]^+$4 transportation-cost inequality, including Gaussian and RDE laws, then there exist constants $[u]^+$5 such that

$[u]^+$6

where $[u]^+$7 and $[u]^+$8 depends on the deviation $[u]^+$9. Solving $[-K,K]$0 provides a quantile bound $[-K,K]$1 and super-uniform p-values

$[-K,K]$2

• Type II Error (Power, Theorem 3.3): For alternatives $[-K,K]$3 with finite first moment,

$[-K,K]$4

with $[-K,K]$5 for $[-K,K]$6, and $[-K,K]$7 the relative entropy. Thus, finite relative entropy ensures nontrivial lower bounds on power.

5. Algorithmic Procedure and Practical Considerations

Population-level smooth-CVaR OC-SVM is implemented via the following high-level steps:

1. Empirical expected signature: $[-K,K]$8
2. Surrogate objective: $[-K,K]$9
3. Joint optimisation: $Q_n(u) = \sum_{i=0}^n a_i u^i$0
   • Solved by alternation or explicit polynomial root-finding when $Q_n(u) = \sum_{i=0}^n a_i u^i$1.
4. Test statistic: $Q_n(u) = \sum_{i=0}^n a_i u^i$2
5. Hypothesis rejection: Reject $Q_n(u) = \sum_{i=0}^n a_i u^i$3 if $Q_n(u) = \sum_{i=0}^n a_i u^i$4.

For sample-based OC-SVM, standard primal/dual QP with signature kernels is used (e.g., LIBSVM, ThunderSVM). At test time, $Q_n(u) = \sum_{i=0}^n a_i u^i$5 is compared to the learned bias $Q_n(u) = \sum_{i=0}^n a_i u^i$6.

6. Empirical Evaluation: Diffusion and Molecular Biology

Anomalous Diffusion

• Setup: Binary discrimination between standard Brownian motion ($Q_n(u) = \sum_{i=0}^n a_i u^i$7) and “spiked-BM” ($Q_n(u) = \sum_{i=0}^n a_i u^i$8) defined by $Q_n(u) = \sum_{i=0}^n a_i u^i$9, $$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$0.
• Statistic: Signature-based distance $$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$1; also, linear form in $$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$2.
• Results:
  • AUROC vs $$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$3: Monotonic increase, no sharp phase at $$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$4.
  • Type I and II control: Empirical p-values ($$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$5 calibration) give marginal FDR $$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$6 but high conditional variability. Weibull tail-bound (Theorem 3.4) with $$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$7 samples yields super-uniform p-values, tighter FDR and FPR control.
  • Comparison: Signature-based distance outperforms TAMSD and is competitive with kernelised OC-SVM.

RNA Modification Detection

• Data: Synthetic 100-nt oligos, three modifications (inosine, m5C, $$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$8) at fixed positions; Nanopore direct RNA reads (Leger et al. 2021).
• Preprocessing: Dorado basecalling, Uncalled4 event alignment, per-base segmentation.
• Methods:
  • OC-SVM on signature features ($$f^n_\alpha(\rho) = \rho + \frac{1}{1-\alpha} E_\mu\left[ Q_n(\langle w, S_N(X) \rangle - \rho) \right], \quad \rho \in [-K, K]$$9, time-augment and invisibility-reset), $(\langle w, S \rangle)^i = \langle w^{\shuffle i}, S \rangle$0 unmodified reads per site, p-values from $(\langle w, S \rangle)^i = \langle w^{\shuffle i}, S \rangle$1 held-out reads.
  • OC-SVM on standard 2D features (mean current and dwell time).
• Results: At BH–FDR level $(\langle w, S \rangle)^i = \langle w^{\shuffle i}, S \rangle$2, signature OC-SVM yields substantially higher recall (power) for all modification types, with type I error controlled at nominal level.

7. Connections, Scope, and Implications

These developments bridge hypothesis testing, path signatures (Lyons et al.), transportation-cost inequalities (Gasteratos and Jacquier 2023), and robust machine learning. The use of smooth CVaR surrogates via shuffle-product identities establishes new analytic techniques for risk calibration and empirical p-value calculation. Non-asymptotic bounds on error rates generalise beyond Gaussian settings to laws of rough differential equation solutions, supporting broader applications in anomalous diffusion analysis and molecular biology. A plausible implication is further cross-fertilisation with time-series anomaly detection and functional data analysis, leveraging closed-form population objectives and signature kernel methods.

The principal contribution is the integration of population-level risk surrogates, shuffle-product algebra, and theoretical guarantees for novelty detection (Gasteratos et al., 2 Dec 2025). This framework enables more refined control of type I and type II errors and supports robust calibration for high-dimensional non-Euclidean data spaces.