The Volterra signature extends the classical path signature by incorporating general matrix-valued kernel into its iterated integral structure, yielding a flexible notion of memory for time series. Its components can be viewed as successive Picard iterates of linear controlled Volterra equations, making their exact computation of additional mathematical interest. However, the kernel introduces substantial algorithmic challenges. We provide a resolution by first decomposing the Chen-type convolution relation established in [13] into analytic and arithmetic parts, and then introducing several efficient algorithms: a general approximative scheme with quadratic complexity O(J2) in the number of time steps J, an FFT-based acceleration with complexity O(J logJ) for convolution kernels on uniform grids, and an exact recursion with complexity O(JR2) for kernels admitting a state-space representation of dimension R; retaining standard signature complexity in the path dimension and truncation level N. We further show that the number of factors in matrix-valued kernels of the form K(t,s) = P p kp(t s)Ap do not increase the asymptotic complexity in J and N. Finally, we derive a finite-difference predictor-corrector scheme for the associated Volterra signature kernel. All algorithms are implemented in the publicly available JAX-based package tensordev.

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this paper, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each item we can recommend is a node and its expected rating is similar to its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret with respect to the optimal policy would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose two algorithms for solving our problem that scale linearly and sublinearly in this dimension. Our experiments on real-world content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens of nodes evaluations.

Ls(d) (30) which is a modification of our supervised loss, whereDl and Du denote the set of labeled and unlabeleddata,respectively.

Projecting this differential equation on the last coordinate givesdHe+1t = dt, that is, He+1t = t. Finally,let (a(n))n N beaCauchysequencein T . Straightforward calculations yield the equality,valid for any x R, tanh(x)=2σ(2x) 1. But,foranyn 1, Next, it is clear that the signature of a constant path is equal to1 = (1,0,...,0,...) which is the nullelementinT . More precisely, fork = 1, C(1;0) = 1 00 = 1 and C(1;1) = 0 01 = 0. Assume that the formula is true at orderk. Then, at order k + 1, there are two cases.

In this paper, we propose to estimate the invariant subspace across heterogeneous multiple networks using a novel bias-corrected joint spectral embedding algorithm. The proposed algorithm recursively calibrates the diagonal bias of the sum of squared network adjacency matrices by leveraging the closed-form bias formula and iteratively updates the subspace estimator using the most recent estimated bias. Correspondingly, we establish a complete recipe for the entrywise subspace estimation theory for the proposed algorithm, including a sharp entrywise subspace perturbation bound and the entrywise eigenvector central limit theorem. Leveraging these results, we settle two multiple network inference problems: the exact community detection in multilayer stochastic block models and the hypothesis testing of the equality of membership profiles in multilayer mixed membership models. Our proof relies on delicate leave-one-out and leave-two-out analyses that are specifically tailored to block-wise symmetric random matrices and a martingale argument that is of fundamental interest for the entrywise eigenvector central limit theorem.

Densest subgraph detection is a fundamental graph mining problem, with a large number of applications. There has been a lot of work on efficient algorithms for finding the densest subgraph in massive networks. However, in many domains, the network is private, and returning a densest subgraph can reveal information about the network. Differential privacy is a powerful framework to handle such settings. We study the densest subgraph problem in the edge privacy model, in which the edges of the graph are private. We present the first sequential and parallel differentially private algorithms for this problem. We show that our algorithms have an additive approximation guarantee. We evaluate our algorithms on a large number of real-world networks, and observe a good privacy-accuracy tradeoff when the network has high density.