Change Detection in Probability Flow ODE: Online Testing in Diffusion Latent Spaces

This paper introduces a distribution-free framework for constructing post-detection confidence sets for changepoints after stopping a sequential change detection procedure. It is well known that conformal test martingales can be used to sequentially detect changes in distribution, but by themselves provide no inference for the time at which a proclaimed change occurred. Past work on post-detection inference requires pre- and post-change classes of distributions to be known, but this paper accomplishes localization of the changepoint without any distributional assumptions. We establish finite-sample coverage guarantees (conditional on correct detection). We provide non-asymptotic bounds on the conditional expected size of the confidence sets. Under suitable asymptotic regimes, we prove that the conditional expected size of the confidence set remains uniformly bounded and demonstrate strong empirical performance on simulated and real data. To the best of our knowledge, this is the first general distribution-free framework for sequential changepoint localization with valid post-detection coverage.

Detecting distributional changes in high dimension is difficult when neither the pre-change nor post-change density is parametrically specified. We introduce a representation-based approach that retains all degree-at-most-two density information while replacing density estimation by matrix mean estimation. For observations in $[-1,1]^d$, a symmetric feature matrix $H_2(X)\in\R^{(d+1)\times(d+1)}$ is constructed so that $M(f)=\E_f H_2(X)$ is an isometric encoding of the degree-two orthogonal projection of the density. We scan matrix CUSUMs after rank-$r$ truncation, exploiting the low rank of the projected jump rather than sparsity of individual coordinates. The resulting \LRD{} estimator has a tent-shaped population objective and a nonasymptotic operator-norm analysis whose leading stochastic term scales as $\sqrt{rd\log(nd)}$. For multiple changes, we give a seeded narrowest-over-threshold procedure and prove exact recovery by an induction that preserves an isolating interval for every undetected change. A cross-fitted scalar refinement learns the changing low-rank direction on one fold and localizes on the other, attaining $\widetilde O_{\Pp}(κ^{-2})$ error; a matching Le Cam lower bound shows optimality up to logarithms. A geometrically $β$-mixing extension follows from a dependent matrix Bernstein inequality. Experiments with ambient dimension up to $200$, a three-change $d=100$ sequence, and a $128$-feature human-activity benchmark show that the method remains computationally practical and accurately detects pure dependence changes that are invisible to mean CUSUMs.

Score-based methods have recently seen increasing popularity in modeling and generation. Methods have been constructed to perform hypothesis testing and change-point detection with score functions, but these methods are in general not as powerful as their likelihood-based peers. Recent works consider generalizing the score-based Fisher divergence into a diffusion-divergence by transforming score functions via multiplication with a matrix-valued function or a weight matrix. In this paper, we extend the score-based hypothesis test and change-point detection stopping rule into their diffusion-based analogs. Additionally, we theoretically quantify the performance of these diffusion-based algorithms and study scenarios where optimal performance is achievable. We propose a method of numerically optimizing the weight matrix and present numerical simulations to illustrate the advantages of diffusion-based algorithms.