arXiv research

Torsion found in knot homology, challenging augmentation theories.

We show how to convert ICL in linearized transformers into model weights.

New conditions ensure Dantzig-Wolfe relaxation matches rank-constrained optimization problems.

Exact solver speeds up Weston-Watkins SVM subproblem significantly.

The paper tackles exact linearization and control of flat discrete-time systems.

Gradient descent achieves exact linear convergence rate for symmetric matrix completion.

This paper analyzes how periodic and soft target updates stabilize linear Q-learning.

Exact posterior score estimation for solving linear inverse problems

Study exact surgery formula in involutive Heegaard Floer homology.

Exact LAD line fitting via PALB with linear scaling and speed.

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact …

Study finds exact limits for sparse regression with fewer observations than usual.

Improved causal discovery methods for large graphs without strict assumptions.

We provide new exact Taylor's series with fixed coefficients and without the remainder. We demonstrate the usefulness of this contribution by using it to obtain very simple solutions to (non-linear) PDEs. We also apply the method to the portfolio model.

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomolog…

Two sweeps of the Brennan-Schwartz algorithm solve American options under negative rates.

MFVI can overestimate predictive variance compared to the exact posterior

We investigate the complexity of deep neural networks (DNN) that represent piecewise linear (PWL) functions. In particular, we study the number of linear regions, i.e. pieces, that a PWL function represented by a DNN can attain, both theoretically and empirically. We present (i) tighter upper and lower bounds for the m…

We develop exact representations of training two-layer neural networks with rectified linear units (ReLUs) in terms of a single convex program with number of variables polynomial in the number of training samples and the number of hidden neurons. Our theory utilizes semi-infinite duality and minimum norm regularization…

A new framework using kernel packets overcomes limitations of state space models for multi-dimensional data.

More and more AI services are provided through APIs on cloud where predictive models are hidden behind APIs. To build trust with users and reduce potential application risk, it is important to interpret how such predictive models hidden behind APIs make their decisions. The biggest challenge of interpreting such predic…

We develop a framework for post model selection inference, via marginal screening, in linear regression. At the core of this framework is a result that characterizes the exact distribution of linear functions of the response $y$, conditional on the model being selected (``condition on selection" framework). This allows…

In this paper, we provide a unified analysis of temporal difference learning algorithms with linear function approximators by exploiting their connections to Markov jump linear systems (MJLS). We tailor the MJLS theory developed in the control community to characterize the exact behaviors of the first and second order …

In this paper, we investigate the non-linear Black--Scholes equation: $u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\quad a,b>0,\ c\geq0.$ and show that the one can be reduced to the equation $u_t+(u_{xx}+u_x)^2=0$ by an appropriate point transformation of variables. For the resulting equation, we study the group-theore…

New method uses diffusion models for Bayesian inverse problems.

The paper develops exact and approximate conformal inference methods for multi-output regression.

A model-based optimal experiment design (OED) of nonlinear systems is studied. OED represents a methodology for optimizing the geometry of the parametric joint-confidence regions (CRs), which are obtained in an a posteriori analysis of the least-squares parameter estimates. The optimal design is achieved by using the a…

Exact minibatch MH method improves scalability for large datasets.

New method solves constrained optimization problems efficiently.

Gaussian processes (GPs) are flexible non-parametric models, with a capacity that grows with the available data. However, computational constraints with standard inference procedures have limited exact GPs to problems with fewer than about ten thousand training points, necessitating approximations for larger datasets. …

We obtain the first positive results for bounded sample compression in the agnostic regression setting with the $\ell_p$ loss, where $p\in [1,\infty]$. We construct a generic approximate sample compression scheme for real-valued function classes exhibiting exponential size in the fat-shattering dimension but independen…

Paper speeds up Gaussian process inference using Matérn kernels.

We analyze the (unconditional) distribution of a linear predictor that is constructed after a data-driven model selection step in a linear regression model. First, we derive the exact finite-sample cumulative distribution function (cdf) of the linear predictor, and a simple approximation to this (complicated) cdf. We t…

Study on dynamics of non-linear autoencoders learning principal components.

Researchers derive exact priors for finite Bayesian neural networks.

Exact causal network discovery is polynomial for sparse networks.

Double descent refers to the phase transition that is exhibited by the generalization error of unregularized learning models when varying the ratio between the number of parameters and the number of training samples. The recent success of highly over-parameterized machine learning models such as deep neural networks ha…

Many applications concern sparse signals, for example, detecting anomalies from the differences between consecutive images taken by surveillance cameras. This paper focuses on the problem of recovering a K-sparse signal x in N dimensions. In the mainstream framework of compressed sensing (CS), the vector x is recovered…

Deep random feature models are analyzed for their performance with exact asymptotic expressions.

Neural network discovers exact solutions to QP with linear constraints.

Exact simulation of correlated binary outcomes using PMF constraints and linear programming.

This work presents an exact solution to the generalized Heston model, where the model parameters are assumed to have linear time dependence The solution for the model in expressed in terms of confluent hypergeometric functions.

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an $m$-by-$n$ nonnegative matrix $X$ and a factorization rank $r$, find, if possible, an $m$-by-$r$ nonnegative matrix $W$ and an $r$-by-$n$ nonnegative matrix $H$ such that $X = WH$. In this paper, we propose two heuristics for exac…

Optimizes financial auditor schedules to reduce time and costs.

This paper considers the problem of estimating the structure of multiple related directed acyclic graph (DAG) models. Building on recent developments in exact estimation of DAGs using integer linear programming (ILP), we present an ILP approach for joint estimation over multiple DAGs, that does not require that the ver…

Study on stable torsion length in groups, showing it vanishes in crystallographic groups and providing algorithms for computation.

In this paper we investigate fiber-wise linear complex Banach sub-Poisson structures defined canonically by the structure of a W*-algebra M. In particular we show that these structures are arranged in the short exact sequence of complex Banach sub-Poisson VB-groupoids with the groupoid of partially invertible elements …

The study shows that the visible range from a point on harmonic manifolds follows an exponential distribution.