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.
Width trees link link invariants and bridge number.
Deep forests enhance expressiveness exponentially with depth, not width or tree size.
Chow and Liu (1968) studied the problem of learning a maximumlikelihood Markov tree. We generalize their work to more complexMarkov networks by considering the problem of learning a maximumlikelihood Markov network of bounded complexity. We discuss howtree-width is in many ways the appropriate measure of complexity and…
We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We use this to give the first examples of knots where any diagram has high tree-widt…
There are many "minimax" complexity functions in mathematics: width of a tree or a link, Heegaard genus of a 3-manifold, the Cheeger constant of a Riemannian manifold. We define such a function w, "width", on countable (or finite) groups and show w(Z^k) = k-1.
We consider compact 3-manifolds M having a submersion h to R in which each generic point inverse is a planar surface. The standard height function on a submanifold of the 3-sphere is a motivating example. To (M, h) we associate a connectivity graph G. For M in the 3-sphere, G is a tree if and only if there is a Fox rei…
We introduce the Mondrian kernel, a fast random feature approximation to the Laplace kernel. It is suitable for both batch and online learning, and admits a fast kernel-width-selection procedure as the random features can be re-used efficiently for all kernel widths. The features are constructed by sampling trees via a…
The paper develops a theory for free boundary minimal surfaces with genus at least one.
This paper uses ML and EVT to analyze tree ring data, improving accuracy of predictions.
Graph homomorphism numbers embed graphs for classification.
In this work we construct a sequence of Riemannian metrics on the three-sphere with scalar curvature greater than or equal to and arbitrarily large widths. Our procedure is based on the connected sum construction of positive scalar curvature metrics due to Gromov and Lawson. We develop analogies between the area of…
This work optimizes neural network bit-width and layer-width for efficiency.
This paper presents a new anytime algorithm for the marginal MAP problem in graphical models. The algorithm is described in detail, its complexity and convergence rate are studied, and relations to previous theoretical results for the problem are discussed. It is shown that the algorithm runs in polynomial-time if the …
New method finds knots without low treewidth diagrams.
We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a time algorithm to compute any Resh…
Many data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies. The approach uses nested stick-breaking processes to allow for trees of unbounded width and depth, where data can live at any node and are infinitely exchangeab…
We show that the class of strongly connected graphical models with treewidth at most k can be properly efficiently PAC-learnt with respect to the Kullback-Leibler Divergence. Previous approaches to this problem, such as those of Chow ([1]), and Ho gen ([7]) have shown that this class is PAC-learnable by reducing it to …
A tree-based dictionary learning model is developed for joint analysis of imagery and associated text. The dictionary learning may be applied directly to the imagery from patches, or to general feature vectors extracted from patches or superpixels (using any existing method for image feature extraction). Each image is …
Paper proposes a VB method for TS-SBP mixture models with reduced computational cost.
This paper finds efficient algorithms for approximating Markov networks with k-tree topologies.
Interactive steering improves hierarchical clustering for diverse user needs.
We present sparse tree-based and list-based density estimation methods for binary/categorical data. Our density estimation models are higher dimensional analogies to variable bin width histograms. In each leaf of the tree (or list), the density is constant, similar to the flat density within the bin of a histogram. His…
New algorithm speeds up knot polynomial calculations.
To infer multilayer deep representations of high-dimensional discrete and nonnegative real vectors, we propose an augmentable gamma belief network (GBN) that factorizes each of its hidden layers into the product of a sparse connection weight matrix and the nonnegative real hidden units of the next layer. The GBN's hidd…
Tree tensor networks balance model complexity and empirical risk for high-dimensional function approximation.
Visualizes classification accuracy and label bias in neural nets and trees.
Given a Gaussian Markov random field, we consider the problem of selecting a subset of variables to observe which minimizes the total expected squared prediction error of the unobserved variables. We first show that finding an exact solution is NP-hard even for a restricted class of Gaussian Markov random fields, calle…
Well-quasi-orders proved on embedded planar graphs.
Decision trees and shallow neural networks have different geometric complexities, impacting their interpretability and accuracy.
We define the Wirtinger width of a knot. Then we prove the Wirtinger width of a knot equals its Gabai width. The algorithmic nature of the Wirtinger width leads to an efficient technique for establishing upper bounds on Gabai width. As an application, we use this technique to calculate the Gabai width of approximately …
Empirical study compares finite- and infinite-width BNNs, revealing performance differences under model mismatch.
The isospectral problem for p-widths is solved using Zoll metrics on S^2.
Lectures on deep learning properties in infinite and large-width networks.
Computed p-widths for hemisphere, first for manifolds with boundary.
Polygon -widths are found via billiard trajectories.
A number of results for C-smooth surfaces of constant width in Euclidean 3-space are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along…
Convolutional Neural Networks (ConvNets) are commonly developed at a fixed resource budget, and then scaled up for better accuracy if more resources are available. In this paper, we systematically study model scaling and identify that carefully balancing network depth, width, and resolution can lead to better performan…
Computed p-widths for real projective plane.
Study bounds Urysohn width of manifolds under surgeries.
Complex captures group properties, invariant under quasi-isometry.
Residual networks with block width max(d_x, d_y) approximate all functions.
While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Ma…
Proves conjecture about sphere widths under rotational symmetry.
New link invariants from diagram colorings match link widths.
Study infinite-depth limits of neural networks with fixed width.
Convolutional neural networks (CNNs) are effective at solving difficult problems like visual recognition, speech recognition and natural language processing. However, performance gain comes at the cost of laborious trial-and-error in designing deeper CNN architectures. In this paper, a genetic programming (GP) framewor…
In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the…
Sharp lower bound for first Neumann eigenvalue found in terms of diameter and width.