arXiv research

Proves exponential sample complexity separations in local differential privacy.

We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This prope…

Mirror flow optimizes separable data problems, converging to a maximum margin classifier.

This study analyzes adversarial training on linearly separable data and finds that gradient updates can achieve large margins in polynomial iterations.

The paper calculates subgroup distortions in 3-manifold groups.

The paper establishes a nearly-sharp statistical threshold for efficient learning in Latent MDPs with separated components.

A random walk $w_n$ on a separable, geodesic hyperbolic metric space $X$ converges to the boundary $\partial X$ with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when …

Analytic curves are classified w.r.t. their symmetry under a regular and separately analytic Lie group action on an analytic manifold. We show that an analytic curve is either exponential or splits into countably many analytic immersive curves, each of them discretely generated by the symmetry group (i.e., each such cu…

We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. We give further Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line b…

We use a new geometric construction, grope splitting, to give a sharp bound for separation of surfaces in 4-manifolds. We also describe applications of this technique in link-homotopy theory, and to the problem of locating pi_1-null surfaces in 4-manifolds. In our applications to link-homotopy, grope splitting serves a…

Higher granularity in MoE models boosts expressivity exponentially.

Let $g \colon S \looparrowright N$ be a properly immersed $π_1$--injective surface in a non-geometric $3$--manifold $N$. We compute the distortion of $π_1(S)$ in $π_1(N)$ and show that how it is related to separability of $π_1(S)$ in $π_1(N)$. The only possibility of the distortion is linear, quadratic, exponential, an…

Memory capacity of DAM scales exponentially with feature separation, unaffected by correlations.

We give a new, effective proof of the separability of cubically convex-cocompact subgroups of special groups. As a consequence, we show that if $G$ is a virtually compact special hyperbolic group, and $Q\leq G$ is a $K$-quasiconvex subgroup, then any $g\in G-Q$ of word-length at most $n$ is separated from $Q$ by a subg…

New methods test discrete distributions faster with local privacy constraints.

Study on hyperbolic groups, focusing on separability and splittings.

Proves depth 2 neural networks can't approximate certain functions as well as depth 3 networks.

The second author previously discussed how classical complexity separation conjectures, we call them "axioms", have implications in three manifold topology: polynomial length stings of operations which preserve certain Jones polynomial evaluations cannot produce exponential simplifications of link diagrams. In this pap…

We develop randomized (block) coordinate descent (CD) methods for linearly constrained convex optimization. Unlike most CD methods, we do not assume the constraints to be separable, but let them be coupled linearly. To our knowledge, ours is the first CD method that allows linear coupling constraints, without making th…

Gradient descent reveals the exact implicit bias via dual optimization for linearly separable data.

New study shows non-interactive privacy model requires exponentially more data.

Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.

Algorithm distinguishes Gaussian mixtures from pure Gaussians in quasi-polynomial time.

Randomly initialized neural networks can linearly separate arbitrary sets.

Paper explains why robust generalization is hard in deep learning models.

New energy functional and fields for Yang-Mills theory, proving monotonicity and vanishing theorems.

Why are classifiers in high dimension vulnerable to "adversarial" perturbations? We show that it is likely not due to information theoretic limitations, but rather it could be due to computational constraints. First we prove that, for a broad set of classification tasks, the mere existence of a robust classifier implie…

We provide a detailed study on the implicit bias of gradient descent when optimizing loss functions with strictly monotone tails, such as the logistic loss, over separable datasets. We look at two basic questions: (a) what are the conditions on the tail of the loss function under which gradient descent converges in the…

New findings on depth vs. width in neural networks, showing depth can improve learnability.

A fast method for training linear classifiers maximizes margins.

Unified view on learning unnormalized distributions using NCE.

Introduces Soft-SVM for binary classification bridging logistic and SVM.

Gradient methods avoid overfitting on separable data.

Mixup reduces the sample complexity of finding optimal decision boundaries for more separable data.

The exponential mechanism is a fundamental tool of Differential Privacy (DP) due to its strong privacy guarantees and flexibility. We study its extension to settings with summaries based on infinite dimensional outputs such as with functional data analysis, shape analysis, and nonparametric statistics. We show that one…

In a previous article, analytic 1-submanifolds had been classified w.r.t. their symmetry under a given regular and separately analytic Lie group action on an analytic manifold. It was shown that such an analytic 1-submanifold is either free or (via the exponential map) analytically diffeomorphic to the unit circle or a…

The past decade has seen substantial work on the use of non-negative matrix factorization and its probabilistic counterparts for audio source separation. Although able to capture audio spectral structure well, these models neglect the non-stationarity and temporal dynamics that are important properties of audio. The re…

We provide new results concerning label efficient, polynomial time, passive and active learning of linear separators. We prove that active learning provides an exponential improvement over PAC (passive) learning of homogeneous linear separators under nearly log-concave distributions. Building on this, we provide a comp…

We adress the maximization problem of expected utility from terminal wealth. The special feature of this paper is that we consider a financial market where the price process of risky assets can have a default time. Using dynamic programming, we characterize the value function with a backward stochastic differential equ…

High-dimensional diffusion models suffer from distorted samples due to CFG.

Hyperbolic space outperforms Euclidean in learning hierarchical data.

We define and discuss the first sparse coding algorithm based on closed-form EM updates and continuous latent variables. The underlying generative model consists of a standard `spike-and-slab' prior and a Gaussian noise model. Closed-form solutions for E- and M-step equations are derived by generalizing probabilistic P…

Study shows depth improves generalization in deep learning models.

In this paper we study perpetual American call and put options in an exponential Lévy model. We consider a negative effective discount rate which arises in a number of financial applications including stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount f…

Study on focal locus of submanifolds in Finsler manifolds, showing regularity and smoothness.

This paper explores robust recovery of a superposition of $R$ distinct complex exponential functions from a few random Gaussian projections. We assume that the signal of interest is of $2N-1$ dimensional and $R<<2N-1$. This framework covers a large class of signals arising from real applications in biology, automation,…

New findings show depth separations for natural radial functions are not possible.

We prove the dimension of any asymptotic cone over a metric space X does not exceed the asymptotic Assouad-Nagata dimension of X. This improves a result of Dranishnikov and Smith who showed that dim(Y) does not exceed asymptotic Assouad-Nagata dimension of X for all separable subsets Y of special asymptotic cones of X …