A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
We introduce a quasi-symmetry invariant of a metric space Z called the capacity dimension. Our main result says that for a visual Gromov hyperbolic space X the asymptotic dimension of X is at most the capacity dimension of its boundary at infinity plus 1.
This paper is devoted to a geometric-measure-theoretic study of the brand new affine BV-capacity which is essentially different from the classic BV-capacity in dimension greater than one.
We prove that every visual Gromov hyperbolic space X whose boundary at infinity has the finite capacity dimension n admits a quasi-isometric embedding into (n+1)-fold product of metric trees.
We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an exa…
We give a bound, linear in the complexity of the surface, on the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.
In this paper we prove a mass-capacity inequality and a volumetric Penrose inequality for conformally flat manifolds, in arbitrary dimensions. As a by-product of the proofs, Pólya-Szegö and Aleksandrov-Fenchel inequalities for mean-convex Euclidean domains are obtained. For each inequality, the case of equality is char…
Proves generalization bounds for SGD using Feller processes and Hausdorff dimension.
problem Characterizing generalization properties of SGD in deep learning.
method Proves generalization bounds for SGD under Feller process approximation, linking generalization error to the Hausdorff dimension of trajectories.
result Generalization error controlled by the Hausdorff dimension of trajectories, which is linked to the tail behavior of the driving process.
Recurrent neural networks are powerful models for processing sequential data, but they are generally plagued by vanishing and exploding gradient problems. Unitary recurrent neural networks (uRNNs), which use unitary recurrence matrices, have recently been proposed as a means to avoid these issues. However, in previous …
This article deals with the generalization performance of margin multi-category classifiers, when minimal learnability hypotheses are made. In that context, the derivation of a guaranteed risk is based on the handling of capacity measures belonging to three main families: Rademacher/Gaussian complexities, metric entrop…
We use the criteria of Lalonde and McDuff to determine a new class of examples of length minimizing paths in the group Ham(M). For a compact symplectic manifold M of dimension two or four, we show that a path in Ham(M), generated by an autonomous Hamiltonian and starting at the identity, which induces no non-cons…
In this paper we study asymptotic behavior of n-superharmonic functions at isolated singularity using the Wolff potential and n-capacity estimates in nonlinear potential theory. Our results are inspired by and extend those of Arsove-Huber and Taliaferro in 2 dimensions. To study n-superharmonic functions we use a…
In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov, and capacity dimension of Buyalo \cite{Buyalo1}) to Nagata-Assouad dimension. This is done by applying two functors on the Lipschitz category of metric spaces: micr…
The study explores how to infer the geometry of space forms from similarity comparisons.
problem Inferring the geometry of space forms from unreliable similarity measurements.
method Introducing ordinal capacity and spread, proving their relation to space form properties, and using statistical analysis of similarity measurements.
result The statistical behavior of ordinal spread variables can identify the underlying space form.
Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. In this paper we unify and generalize these strategies by viewing any geodesic metric space as a countable union of stable subspaces: we show that every stable subgroup is a qu…
Vapnik-Chervonenkis (VC) dimension is a fundamental measure of the generalization capacity of learning algorithms. However, apart from a few special cases, it is hard or impossible to calculate analytically. Vapnik et al. [10] proposed a technique for estimating the VC dimension empirically. While their approach behave…
We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the to…
We study finite sample expressivity, i.e., memorization power of ReLU networks. Recent results require N hidden nodes to memorize/interpolate arbitrary N data points. In contrast, by exploiting depth, we show that 3-layer ReLU networks with Ω(N) hidden nodes can perfectly memorize most datasets with N po…
We study various capacities on compact Kähler manifolds which generalize the Bedford-Taylor Monge-Ampère capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Ampère equations.
Working in high-dimensional latent spaces, the internal encoding of data in Variational Autoencoders becomes naturally sparse. We discuss this known but controversial phenomenon sometimes refereed to as overpruning, to emphasize the under-use of the model capacity. In fact, it is an important form of self-regularizatio…
In this article, we propose the notion of the general p-affine capacity and prove some basic properties for the general p-affine capacity, such as affine invariance and monotonicity. The newly proposed general p-affine capacity is compared with several classical geometric quantities, e.g., the volume, the p-var…
While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of sympl…
Study excess capacity in neural networks using Rademacher complexity.
problem Understanding how much capacity deep networks have beyond what's needed for classification.
method Unified Rademacher complexity bounds for function composition and convolutional layers, considering Lipschitz constants and initialization norms.
result There is substantial excess capacity per task, and capacity can be kept similar across different tasks.
Study capacity constraints in continual learning with a simple model.
problem Understanding optimal resource allocation for agents with limited memory and compute resources.
method Analyzes a capacity-constrained linear-quadratic-Gaussian (LQG) sequential prediction problem and demonstrates optimal capacity allocation strategies.
result Derives a solution to the capacity-constrained LQG sequential prediction problem and shows how to optimally allocate capacity across sub-problems in the steady state.