4 articles · sorted by date
When data live on a curved surface, ordinary denoising score matching can get tangled in a noise channel whose variance blows up like d/σ^2 as the corruption level σ shrinks. This paper shows a clean way out: condition on the nearest-point projection π(X), and the singularity disappears. The resulting conditional expectation is the unique L2-optimal Rao-Blackwellized predictor among all estimators that depend only on the projected observation. The authors then expand this canonical target for small noise and show that it matches the intrinsic Riemannian score up to an explicit order-σ^2 correction. That correction splits into an intrinsic Tweedie term and an extrinsic curvature term involving the Weingarten and Ricci operators. In the flat case, the method becomes ordinary lower-dimensional Gaussian denoising score matching exactly. On the sphere S^d, the extrinsic piece simplifies to (1−d/2)∇_M log q, and it cancels on S^2 even though the intrinsic Tweedie term remains.
When a diversity model is forced to pick exactly k items, its hidden parameters become harder to pin down. This paper studies determinantal point processes through a spectral decomposition, where eigenvalues control how many items appear and the eigenvector directions control which items appear together. For the fixed-size version, called a k-DPP, the authors show that the spectral part is identifiable only up to a common scale, and the orientation part is recoverable only through squared minors of the eigenvector matrix. They describe the identifiability gap with three explicit symmetries: scale, sign similarity, and eigenspace rotation. They also prove a dimension-counting theorem: whenever binom(N,k) is smaller than N(N+1)/2, additional continuous non-identifiability must exist. In the full DPP, by contrast, the only non-identifiability comes from a discrete sign similarity.
If you remove the bad data points from a time series, you might expect the analysis to recover the truth. This paper shows that expectation can fail in dynamic time series models. Even when the contamination locations are known perfectly, deleting the contaminated observations does not restore the clean-data objective, because the contamination keeps spreading through the residual filter and warps the estimation criterion. The result is stark: subsample-based estimators are generically inconsistent for the parameter of the uncontaminated process. The authors describe this as a structural clash between pointwise subsampling and residual propagation. To fix it, they introduce a patch removal operator, a way of transforming index sets so the estimator respects how contamination moves through the model. Under general high-level conditions, this new transformation leaves the estimator asymptotically unchanged when the data are clean, while restoring consistency under contamination. The approach applies to a broad class of residual-based estimators and does not require modelling the contamination process.
If a decision-making algorithm makes a small mistake today, how far can that error travel by the end of the problem? This paper studies that question in discrete-time stochastic optimal control, where choices are made step by step under uncertainty. The authors estimate the value function — the number that tells you how good each choice is — by combining kernel ridge regression, a nonparametric regression method in a reproducing kernel Hilbert space (RKHS), with Monte Carlo subsampling for the continuation value, meaning the payoff expected if you keep going. They then build an error decomposition and control the error terms at each time step. The central result is an analysis of how those errors propagate backward in time, from maturity to the initial stage, which the paper says is relatively underexplored in stochastic control. The same framework is then applied to American option pricing, showing how a theory of backward error can speak directly to a major financial problem.
