7,930 research outputs found
The MacWilliams Identity for Krawtchouk Association Schemes
The weight distribution of an error correcting code is a crucial statistic in determining its performance. One key tool for relating the weight of a code to that of its dual is the MacWilliams Identity, first developed for the Hamming association scheme. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via eigenvalues of the association scheme. The functional transformation form can, in particular, be used to derive important moment identities for the weight distribution of codes. In this thesis, we focus initially on extending the functional transformation to codes based on skew-symmetric and Hermitian matrices. A generalised b-algebra and new fundamental homogeneous polynomials are then identified and proven to generate the eigenvalues of a specific subclass of association schemes, Krawtchouk association schemes. Based on the new set of MacWilliams Identities as a functional transform, we derive several moments of the weight distribution for all of these codes
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
The Entropic journey of Kac's Model
The goal of this paper is to review the advances that were made during the
last few decades in the study of the entropy, and in particular the entropy
method, for Kac's many particle system.Comment: This paper has been written to commemorate Maria Concei\c{c}\~{a}o
Carvalho (S\~{a}o), who will always be with us in our memories and heart
Counterfactual Causality for Reachability and Safety based on Distance Functions
Investigations of causality in operational systems aim at providing
human-understandable explanations of why a system behaves as it does. There is,
in particular, a demand to explain what went wrong on a given counterexample
execution that shows that a system does not satisfy a given specification. To
this end, this paper investigates a notion of counterfactual causality in
transition systems based on Stalnaker's and Lewis' semantics of counterfactuals
in terms of most similar possible worlds and introduces a novel corresponding
notion of counterfactual causality in two-player games. Using distance
functions between paths in transition systems, this notion defines whether
reaching a certain set of states is a cause for the violation of a reachability
or safety property. Similarly, using distance functions between memoryless
strategies in reachability and safety games, it is defined whether reaching a
set of states is a cause for the fact that a given strategy for the player
under investigation is losing. The contribution of the paper is two-fold: In
transition systems, it is shown that counterfactual causality can be checked in
polynomial time for three prominent distance functions between paths. In
two-player games, the introduced notion of counterfactual causality is shown to
be checkable in polynomial time for two natural distance functions between
memoryless strategies. Further, a notion of explanation that can be extracted
from a counterfactual cause and that pinpoints changes to be made to the given
strategy in order to transform it into a winning strategy is defined. For the
two distance functions under consideration, the problem to decide whether such
an explanation imposes only minimal necessary changes to the given strategy
with respect to the used distance function turns out to be coNP-complete and
not to be solvable in polynomial time if P is not equal to NP, respectively.Comment: This is the extended version of a paper accepted for publication at
GandALF 202
When Deep Learning Meets Polyhedral Theory: A Survey
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 of neural networks converged back to simpler representations
based on piecewise constant and piecewise linear functions such as the
Rectified Linear Unit (ReLU), which became the most commonly used type of
activation function in neural networks. That made certain types of network
structure \unicode{x2014}such as the typical fully-connected feedforward
neural network\unicode{x2014} amenable to analysis through polyhedral theory
and to the application of methodologies such as Linear Programming (LP) and
Mixed-Integer Linear Programming (MILP) for a variety of purposes. In this
paper, we survey the main topics emerging from this fast-paced area of work,
which bring a fresh perspective to understanding neural networks in more detail
as well as to applying linear optimization techniques to train, verify, and
reduce the size of such networks
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