arXiv research

We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allows one to systematically develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. The approach generalizes the notion o…

A new method approximates the Sliced-Wasserstein distance without random projections.

We propose fast approximations for the generalized sliced-Wasserstein distance.

Develops optimal low-dimensional approximations to high-dimensional SDEs.

New method reduces computational cost for nonnegative low rank matrix approximation.

Extends Tian theorem to Vaisman manifolds for approximations.

Improved semialgebraic choices with linear complexity.

Paper analyzes LPSA algorithm for constrained optimization, revealing phase transitions and bias-variance trade-offs.

This paper surveys various methods for dimensionality reduction and nearest neighbor search.

Solves Yau-Tian-Donaldson conjecture for smooth projective varieties.

Approximate inference via information projection has been recently introduced as a general-purpose approach for efficient probabilistic inference given sparse variables. This manuscript goes beyond classical sparsity by proposing efficient algorithms for approximate inference via information projection that are applica…

A novel PP algorithm using GMMs and GAs for detecting informative structures.

In this paper, we study asymptotic behavior of projective embeddings of Kummer varieties given by theta functions, and their amoebas. We prove that a Lagrangian fibration of the Kummer variety can be approximated by moment maps of the projective spaces.

Policy evaluation with linear function approximation is an important problem in reinforcement learning. When facing high-dimensional feature spaces, such a problem becomes extremely hard considering the computation efficiency and quality of approximations. We propose a new algorithm, LSTD($λ$)-RP, which leverages rando…

We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoiding explicit principal …

Let $P : Σ\rightarrow S$ be a finite degree covering map between surfaces. Rafi and Schleimer show that there is an induced quasi-isometric embedding $Π: \mathcal{C}(S) \rightarrow \mathcal{C}(Σ)$ between the associated curve complexes. We define an operation on curves in $\mathcal{C}(Σ)$ using minimal intersection num…

Study on volumes of random inscribed polytopes in projective geometries.

The study explores holomorphic Legendrian curves and superminimal surfaces in complex projective and sphere spaces.

Soft-Radial Projection solves gradient saturation in constrained deep learning.

Projection-cost preservation is a low-rank approximation guarantee which ensures that the cost of any rank-$k$ projection can be preserved using a smaller sketch of the original data matrix. We present a general structural result outlining four sufficient conditions to achieve projection-cost preservation. These condit…

Paper optimizes approximating high-dimensional diffusions by independent coordinates.

Low-rank structure have been profoundly studied in data mining and machine learning. In this paper, we show a dense matrix $X$'s low-rank approximation can be rapidly built from its left and right random projections $Y_1=XA_1$ and $Y_2=X^TA_2$, or bilateral random projection (BRP). We then show power scheme can further…

This paper deals with sparse feature selection and grouping for classification and regression. The classification or regression problems under consideration consists in minimizing a convex empirical risk function subject to an $\ell^1$ constraint, a pairwise $\ell^\infty$ constraint, or a pairwise $\ell^1$ constraint. …

We study the use of "sign $α$-stable random projections" (where $0<α\leq 2$) for building basic data processing tools in the context of large-scale machine learning applications (e.g., classification, regression, clustering, and near-neighbor search). After the processing by sign stable random projections, the inner pr…

We address the problem of automatic generation of features for value function approximation. Bellman Error Basis Functions (BEBFs) have been shown to improve the error of policy evaluation with function approximation, with a convergence rate similar to that of value iteration. We propose a simple, fast and robust algor…

Stochastic approximation algorithms show exponential progress bounds.

New explanation of reservoir computing using random projections.

MPE framework proves universal approximation for quantum data distribution.

Develops precise expressions for random projections for better machine learning tasks.

Unified theory and debiasing framework for random oblique projections in high dimensions.

Stochastic differential equation approximation for linear TD(0) under Markovian noise

We introduce cylindrical projections to simulate infinite-dimensional occupation flows of diffusions.

Develops an oblique projection technique to approximate a foliation for non-normal dynamics.

The paper interprets diffusion models as gradient descent and proposes a new sampler.

Constructs stable Hilbert bundles on curves using Diophantine approximation.

Paper studies binary random projections with controllable sparsity patterns for computational and accuracy advantages.

A new method for Bayesian inference tackles high-dimensional problems.

Unified framework for multi-view learning with orthogonal projections.

Bayesian deep learning avoids underfitting by projecting onto null space of generalized Gauss-Newton matrix.

We study the problem of determining the optimal low dimensional projection for maximising the separability of a binary partition of an unlabelled dataset, as measured by spectral graph theory. This is achieved by finding projections which minimise the second eigenvalue of the graph Laplacian of the projected data, whic…

Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on how many projections are needed to accurately preserve the geometry of these man…

The method of random projections has become a standard tool for machine learning, data mining, and search with massive data at Web scale. The effective use of random projections requires efficient coding schemes for quantizing (real-valued) projected data into integers. In this paper, we focus on a simple 2-bit coding …

The paper projects unknown manifolds onto hyperspheres for efficient function approximation.

Random projections simplify complex data for classification.

We show that the loop spaces of real projective spaces are topologically approximated by the spaces of rational maps from RP(1) to RP(n). As a byproduct of our constructions we obtain an interpretation of the Kronecker characteristic (degree) of an ornament via particle spaces.

For every fibration $f : X \to B$ with $X$ a compact Kähler manifold, $B$ a smooth projective curve, and a general fiber of $f$ an abelian variety, we prove that $f$ has an algebraic approximation.

This paper is on the normal approximation of singular subspaces when the noise matrix has i.i.d. entries. Our contributions are three-fold. First, we derive an explicit representation formula of the empirical spectral projectors. The formula is neat and holds for deterministic matrix perturbations. Second, we calculate…

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is c…