arXiv research

Study shows different fundamental groups for manifolds with same limit.

We show through case studies that it is easier to estimate the fundamental limits of data processing than to construct explicit algorithms to achieve those limits. Focusing on binary classification, data compression, and prediction under logarithmic loss, we show that in the finite space setting, when it is possible to…

In this note we investigate to what extent the fundamental group of a metric space can be described as the inverse limit of its discrete fundamental groups. We show that some mild conditions suffice to imply the existence of an isomorphism and we provide a list of counterexamples to possible weakenings of these hypothe…

In this paper, we examine the fundamental performance limits of prediction, with or without side information. More specifically, we derive generic lower bounds on the $\mathcal{L}_p$ norms of the prediction errors that are valid for any prediction algorithms and for any data distributions. Meanwhile, we combine the ent…

Study shows a central limit theorem for random coverings of manifolds with nilpotent groups.

This work characterizes the fundamental limit of network pruning using statistical dimension and convex geometry.

Study on matching nodes between graphs to preserve edges, focusing on limits and algorithms.

Paper explores limits of exact inference in structured prediction models.

In this paper, we utilize information theory to study the fundamental performance limitations of generic feedback systems, where both the controller and the plant may be any causal functions/mappings while the disturbance can be with any distributions. More specifically, we obtain fundamental $\mathcal{L}_p$ bounds on …

Characterizes fundamental groups of disjointly tree-graded spaces.

Study on the limits of projective special real manifolds and their symmetries.

We study the limits and methods of training two-layer autoencoders.

We study direct limits of embedded Cantor sets and embedded \sier curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called $ω$-Cantor spaces, and similarly, all limits of \sier curves give homeomorphic spaces, called to $ω$-\sier curves. W…

This paper sets fundamental limits for rank-one matrix estimation with varying noise levels.

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a regular domain G of R^3. We prove that the second fundamental form of the boundary surface of G is nonnegatively defined at every point of the limit set of such immersions.

Paper proposes MMC to avoid high-density bias in clustering.

When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give…

A new method combines simple binary classifiers to build complex multiclass classifiers, achieving performance limits in a Gaussian setting.

The fundamental group of a hyperbolic manifold acts on the limit set, giving rise to a cross-product C^* algebra. We construct nontrivial K-cycles for the cross-product algebra, thereby extending some results of Connes and Sullivan to higher dimensions. We also show how the Patterson-Sullivan measure on the limit set c…

New research limits how well attackers can guess if data points were in a model's training set.

In this note we establish several versions of a compactness theorem for submanifolds. In particular we require only bounds on the second fundamental form and do not assume volume or diameter bounds. As an application we prove a compactness theorem for mean curvature flows and use it to construct smooth blow-up limits a…

The paper studies random covers of torus knot complements and their statistical properties.

Turnover-adjusted IR is always lower than classic IR, suggesting managers can improve performance by limiting turnover.

Esnault asked whether every smooth complex projective variety with infinite fundamental group has a nonzero symmetric differential (a section of a symmetric power of the cotangent bundle). In a sense, this would mean that every variety with infinite fundamental group has some nonpositive curvature. We show that the ans…

Endowed with natural topologies, the fundamental group of the Hawaiian earring continuously injects into the inverse limit of free groups. This note shows the injection fails to have a continuous inverse. Such a phenomenon was unexpected and appears to contradict results of another author.

In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals. Also, we characterize in a simple way these singular surfaces and its fundament…

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I $κ$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's pr…

The paper sets fundamental limits for ERM in high dimensions.

The paper characterizes the efficiency of transferring knowledge from a teacher to a student classifier over finite domains.

By using Thurston's bending construction we obtain a sequence of faithful discrete representations ρ_n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic space H^4. The algebraic limit of ρ_n contains a finitely generated subgroup F whos…

In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in $B_k\times (0,T)$ to the minimal fundamental solut…

This work examines fundamental limits in model falsification without assuming specific distributions.

We explore the limit set of a particular spherical CR uniformization of a cusped hyperbolic manifold. We prove that the limit set is the closure of a countable union of $\mathbb{R}$-circles, is connected, and contains a Hopf link with three components; we also show that the fundamental group of its complement in $S^3$ …

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

We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In particular, we derive the existence of hyperbolic knot complements which contain ba…

Paper models and compresses wideband CSI feedback in FDD MIMO systems.

Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.

Study limits of curved spaces with boundaries.

We consider the possible Euler characteristics and fundamental groups of the complementary components $X$ and $Y$ of an embedding of a connected closed 3-manifold $M$ in $S^4$. We use a 2-knot satellite construction to change the fundamental groups, and Massey products to limit the values of $χ(X)$ and $χ(Y)$ when $M$ …

Paper analyzes online reinforcement learning with outcome-based feedback, providing efficient algorithms and fundamental limits.

Optimal first-order methods are shown to be fundamental limits in functional estimation.

The paper explores game-theoretic alignment of LLMs with human preferences, finding limitations and conditions.

We give an expository account of our proof that each cusp-free hyperbolic 3-manifold M with finitely generated fundamental group and incompressible ends is an algebraic limit of geometrically finite hyperbolic 3-manifolds.

Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of infinite index. As a corollary, the same holds for limit groups. We also obtain a ch…

Paper studies fundamental limits of communication in distributed learning.

Approximate inference algorithm is one of the fundamental research fields in machine learning. The two dominant theoretical inference frameworks in machine learning are variational inference (VI) and Markov chain Monte Carlo (MCMC). However, because of the fundamental limitation in the theory, it is very challenging to…

Study on the geometry of limit spaces of manifolds with boundary.

In this paper, we obtain fundamental $\mathcal{L}_{p}$ bounds in sequential prediction and recursive algorithms via an entropic analysis. Both classes of problems are examined by investigating the underlying entropic relationships of the data and/or noises involved, and the derived lower bounds may all be quantified in…