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Convolution is an efficient technique to obtain abstract feature representations using hierarchical layers in deep networks. Although performing convolution in Euclidean geometries is fairly straightforward, its extension to other…
High-multiplicity signatures at particle colliders can arise in Standard Model processes and beyond. With such signatures, difficulties often arise from the large dimensionality of the kinematic space. For final states containing a single…
We investigate two optimization problems on area-proportional rectangle contact representations for layered, embedded planar graphs. The vertices are represented as interior-disjoint unit-height rectangles of prescribed widths, grouped in…
A number of recent studies have shown that a Deep Convolutional Neural Network (DCNN) pretrained on a large dataset can be adopted as a universal image description which leads to astounding performance in many visual classification tasks.…
A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by…
We introduce general scattering transforms as mathematical models of deep neural networks with l2 pooling. Scattering networks iteratively apply complex valued unitary operators, and the pooling is performed by a complex modulus. An…
In this paper, we investigate the geometric structure of activation spaces of fully connected layers in neural networks and then show applications of this study. We propose an efficient approximation algorithm to characterize the convex…
Characterization of entropy functions is of fundamental importance in information theory. By imposing constraints on their Shannon outer bound, i.e., the polymatroidal region, one obtains the faces of the region and entropy functions on…
In the past decade, deep learning became the prevalent methodology for predictive modeling thanks to the remarkable accuracy of deep neural networks in tasks such as computer vision and natural language processing. Meanwhile, the structure…
Mathematical psychology has a long tradition of modeling probabilistic choice via distribution-free random utility models and associated random preference models. For such models, the predicted choice probabilities often form a bounded and…
We study the expressibility and learnability of convex optimization solution functions and their multi-layer architectural extension. The main results are: \emph{(1)} the class of solution functions of linear programming (LP) and quadratic…
Multilayer network analysis has become a vital tool for understanding different relationships and their interactions in a complex system, where each layer in a multilayer network depicts the topological structure of a group of nodes…
This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of…
Sliding window sums are widely used for string indexing, hashing and time series analysis. We have developed a family of the generic vectorized sliding sum algorithms that provide speedup of O(P/w) for window size $w$ and number of…
The monotone path polytope of a polytope $P$ encapsulates the combinatorial behavior of the shadow vertex rule (a pivot rule used in linear programming) on $P$. Computing monotone path polytopes is the entry door to the larger subject of…
Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of…
Symmetric functions, which take as input an unordered, fixed-size set, are known to be universally representable by neural networks that enforce permutation invariance. These architectures only give guarantees for fixed input sizes, yet in…
We introduce the M-representation of polytopes, which makes it possible to compute linear transformations, convex hulls, and Minkowski sums with linear complexity in the dimension of the polytopes. When the polytope is a convex hull of a…
This article demonstrates that convolutional operation can be converted to matrix multiplication, which has the same calculation way with fully connected layer. The article is helpful for the beginners of the neural network to understand…
Bisztriczky defines a multiplex as a generalization of a simplex, and an ordinary polytope as a generalization of a cyclic polytope. This paper presents results concerning the combinatorics of multiplexes and ordinary polytopes. The flag…