arXiv research

Algorithm learns weight matrix from single trajectory of nonlinear dynamical system.

Trajectory owner prediction is the basis for many applications such as personalized recommendation, urban planning. Although much effort has been put on this topic, the results archived are still not good enough. Existing methods mainly employ RNNs to model trajectories semantically due to the inherent sequential attri…

Paper proposes a new descriptor for early trajectory characterization in matrix iterations.

Study on overlaps of singular vectors in Gaussian matrix submatrices.

Paper studies asymmetric matrix sensing, proving gradient descent converges to low-rank solutions.

The paper explores states of financial markets using correlation matrices and their dynamics.

Method estimates number of clusters in Block Markov Chain trajectories.

OMD monitors stock market dynamics through matrix trajectories, revealing crisis patterns and sector rotations.

Gradient Descent with small random initialization solves rank-1 matrix completion efficiently.

Gradient descent with small random init mimics spectral methods for low-rank matrix recovery.

GD with large init shows incremental learning in matrix factorization.

New method predicts and optimizes matrix recovery from noisy measurements.

We study the problem of identity testing of markov chains. In this setting, we are given access to a single trajectory from a markov chain with unknown transition matrix $Q$ and the goal is to determine whether $Q = P$ for some known matrix $P$ or $\text{Dist}(P, Q) \geq ε$ where $\text{Dist}$ is suitably defined. In r…

NoTMF forecasts sparse urban road movement speeds with nonstationary temporal matrix factorization.

Study on estimating unstable open-loop matrices from state trajectories.

A new algorithm for offline RL with trajectory-wise reward reduces bias and variance errors.

Effective Gram matrix predicts deep network generalization.

We consider a strictly pathwise setting for Delta hedging exotic options, based on Föllmer's pathwise Itō calculus. Price trajectories are $d$-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix. The existence of Delta hedging strategie…

We consider cones over manifolds admitting real Killing spinors and instanton equations on connections on vector bundles over these manifolds. Such cones are manifolds with special (reduced) holonomy. We generalize the scalar ansatz for a connection proposed by Harland and Nolle in such a way that instantons are parame…

New spectral methods improve matrix estimation in RL with low-rank structure.

Identifies latent actions and dynamics from offline data with diverse demonstrators.

The extended Kalman filter is perhaps the most standard tool to estimate in real time the state of a dynamical system from noisy measurements of some function of the system, with extensive practical applications (such as position tracking via GPS). While the plain Kalman filter for linear systems is well-understood, th…

ATLAS adapts HMC step size and trajectory length for complex geometries.

Ensemble clustering has been a popular research topic in data mining and machine learning. Despite its significant progress in recent years, there are still two challenging issues in the current ensemble clustering research. First, most of the existing algorithms tend to investigate the ensemble information at the obje…

In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the cost function. The first proposed problem is control for positive definite Hermiti…

We introduce a probabilistic generative model for disentangling spatio-temporal disease trajectories from series of high-dimensional brain images. The model is based on spatio-temporal matrix factorization, where inference on the sources is constrained by anatomically plausible statistical priors. To model realistic tr…

Framework for joint inference of network topology and interaction types in heterogeneous systems.

The annihilating filter-based low-rank Hankel matrix approach (ALOHA) is one of the state-of-the-art compressed sensing approaches that directly interpolates the missing k-space data using low-rank Hankel matrix completion. The success of ALOHA is due to the concise signal representation in the k-space domain thanks to…

Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g. trimming, regularized cost, projection) in order…

Matrix Chernoff bound for Markov chains applied to co-occurrence matrices.

The paper validates a method for recovering over-parameterized matrices and images from noisy measurements.

Theoretical analysis of the error landscape of deep neural networks has garnered significant interest in recent years. In this work, we theoretically study the importance of noise in the trajectories of gradient descent towards optimal solutions in multi-layer neural networks. We show that adding noise (in different wa…

Empirical data on real complex systems are becoming increasingly available. Parallel to this is the need for new methods of reconstructing (inferring) the topology of networks from time-resolved observations of their node-dynamics. The methods based on physical insights often rely on strong assumptions about the proper…

Although many successful ensemble clustering approaches have been developed in recent years, there are still two limitations to most of the existing approaches. First, they mostly overlook the issue of uncertain links, which may mislead the overall consensus process. Second, they generally lack the ability to incorpora…

This work extends entropic optimal transport to non-product reference couplings, focusing on Gaussian cases.

The paper tackles transfer learning for growing matrix representations, improving estimation accuracy.

StrTransformer recovers sources without labels by optimizing latent matrices and enforcing structural constraints.

This paper proposes a new approach to describe the stability of linear time-invariant systems via the torsion $τ(t)$ of the state trajectory. For a system $\dot{r}(t)=Ar(t)$ where $A$ is invertible, we show that (1) if there exists a measurable set $E_1$ with positive Lebesgue measure, such that $r(0)\in E_1$ implies t…

Study non-asymptotic bounds on correlation in high-dimensional linear systems, revealing invariant subspaces and bottlenecks.

Diffusion models' consistency across splits explained by random matrix theory.

Stochastic gradient descent (SGD) is a key ingredient in the training of deep neural networks and yet its geometrical significance appears elusive. We study a deterministic model in which the trajectories of our dynamical systems are described via geodesics of a family of metrics arising from the diffusion matrix. Thes…

Matrix factorization is a key component of collaborative filtering-based recommendation systems because it allows us to complete sparse user-by-item ratings matrices under a low-rank assumption that encodes the belief that similar users give similar ratings and that similar items garner similar ratings. This paradigm h…

We study the problem of discriminative sub-trajectory mining. Given two groups of trajectories, the goal of this problem is to extract moving patterns in the form of sub-trajectories which are more similar to sub-trajectories of one group and less similar to those of the other. We propose a new method called Statistica…

This paper studies the estimation of low-rank Markov chains from empirical trajectories. We propose a non-convex estimator based on rank-constrained likelihood maximization. Statistical upper bounds are provided for the Kullback-Leiber divergence and the $\ell_2$ risk between the estimator and the true transition matri…

We address the problem of estimating the mixing time of a Markov chain from a single trajectory of observations. Unlike most previous works which employed Hilbert space methods to estimate spectral gaps, we opt for an approach based on contraction with respect to total variation. Specifically, we estimate the contracti…

New methods learn sampling distributions for particle filters without supervision.

Paper uses deep imitation learning to predict aircraft trajectories accurately.

One way to avoid overfitting in machine learning is to use model parameters distributed according to a Bayesian posterior given the data, rather than the maximum likelihood estimator. Stochastic gradient Langevin dynamics (SGLD) is one algorithm to approximate such Bayesian posteriors for large models and datasets. SGL…