33,161 research outputs found
Variations of Independence in Boolean Algebras
We give a definition of some classes of boolean algebras generalizing free
boolean algebras; they satisfy a universal property that certain functions
extend to homomorphisms. We give a combinatorial property of generating sets of
these algebras, which we call n -independent. The properties of these classes
(n-free and omega-free boolean algebras) are investigated. These include
connections to hypergraph theory and cardinal invariants on these algebras.
Related cardinal functions, n Ind, which is the supremum of the cardinalities
of n-independent subsets; i_n, the minimum size of a maximal n -independent
subset; and i_omega, the minimum size of an omega-independent subset, are
introduced and investigated. The values of i_n and i_omega on P(omega)/fin are
shown to be independent of ZFC. Ideal-independence is also considered, and it
is shown that the cardinal function p <= s_mm for infinite boolean algebras. We
also define and consider moderately generated boolean algebras; that is, those
boolean algebras that have a generating set consisting of elements that split
finitely many elements of the boolean algebra.Comment: PhD thesi
Monotone Boolean Functions, Feasibility/Infeasibility, LP-type problems and MaxCon
This paper outlines connections between Monotone Boolean Functions, LP-Type
problems and the Maximum Consensus Problem. The latter refers to a particular
type of robust fitting characterisation, popular in Computer Vision (MaxCon).
Indeed, this is our main motivation but we believe the results of the study of
these connections are more widely applicable to LP-type problems (at least
'thresholded versions', as we describe), and perhaps even more widely. We
illustrate, with examples from Computer Vision, how the resulting perspectives
suggest new algorithms. Indeed, we focus, in the experimental part, on how the
Influence (a property of Boolean Functions that takes on a special form if the
function is Monotone) can guide a search for the MaxCon solution.Comment: Parts under conference review, work in progress. Keywords: Monotone
Boolean Functions, Consensus Maximisation, LP-Type Problem, Computer Vision,
Robust Fitting, Matroid, Simplicial Complex, Independence System
Non-Commutative Stochastic Independence and Cumulants
In a central lemma we characterize "generating functions" of certain functors
on the category of algebraic non-commutative probability spaces. Special
families of such generating functions correspond to "unital, associative
universal products" on this category, which again define a notion of
non-commutative stochastic independence. Using the central lemma, we can prove
the existence of cumulants and of "cumulant Lie-algebras" for a wide class of
independences. These include the five independences (tensor, free, Boolean,
monotone, anti-monotone) appearing in N. Murakis classification, c-free
independence of M. Bozejko and R. Speicher, the indented product of T. Hasebe
and the bi-free independence of D. Voiculescu. We show that the non-commutative
independence can be reconstructed from its cumulants and cumulant Lie algebras
Topological Perspectives on Statistical Quantities I
In statistics cumulants are defined to be functions that measure the linear
independence of random variables. In the non-communicative case the Boolean
cumulants can be described as functions that measure deviation of a map between
algebras from being an algebra morphism. In Algebraic topology maps that are
homotopic to being algebra morphisms are studied using the theory of
algebras. In this paper we will explore the link between these two points of
views on maps between algebras that are not algebra maps
Polynomial Threshold Functions: Structure, Approximation and Pseudorandomness
We study the computational power of polynomial threshold functions, that is,
threshold functions of real polynomials over the boolean cube. We provide two
new results bounding the computational power of this model.
Our first result shows that low-degree polynomial threshold functions cannot
approximate any function with many influential variables. We provide a couple
of examples where this technique yields tight approximation bounds.
Our second result relates to constructing pseudorandom generators fooling
low-degree polynomial threshold functions. This problem has received attention
recently, where Diakonikolas et al proved that -wise independence suffices
to fool linear threshold functions. We prove that any low-degree polynomial
threshold function, which can be represented as a function of a small number of
linear threshold functions, can also be fooled by -wise independence. We
view this as an important step towards fooling general polynomial threshold
functions, and we discuss a plausible approach achieving this goal based on our
techniques.
Our results combine tools from real approximation theory, hyper-contractive
inequalities and probabilistic methods. In particular, we develop several new
tools in approximation theory which may be of independent interest
Relations between convolutions and transforms in operator-valued free probability
We introduce a class of independence relations, which include free, Boolean
and monotone independence, in operator valued probability. We show that this
class of independence relations have a matricial extension property so that we
can easily study their associated convolutions via Voiculescu's fully matricial
function theory. Based the matricial extension property, we show that many
results can be generalized to multi-variable cases. Besides free, Boolean and
monotone independence convolutions, we will focus on two important
convolutions, which are orthogonal and subordination additive convolutions. We
show that the operator-valued subordination functions, which come from the free
additive convolutions or the operator-valued free convolution powers, are
reciprocal Cauchy transforms of operator-valued random variables which are
uniquely determined up to Voiculescu's fully matricial function theory. In the
end, we study relations between certain convolutions and transforms in
-operator valued probability.Comment: Some typos are corrected. Comments welcom
Domain Reduction for Monotonicity Testing: A Tester for Boolean Functions in -Dimensions
We describe a -query monotonicity tester for Boolean
functions on the -hypergrid. This is the first
monotonicity tester with query complexity independent of . Motivated by this
independence of , we initiate the study of monotonicity testing of
measurable Boolean functions over the continuous
domain, where the distance is measured with respect to a product distribution
over . We give a -query monotonicity tester
for such functions.
Our main technical result is a domain reduction theorem for monotonicity. For
any function , let be its distance to
monotonicity. Consider the restriction of the function on a random
sub-hypergrid of the original domain. We show that for , the expected distance of the restriction is
. Previously, such a
result was only known for (Berman-Raskhodnikova-Yaroslavtsev, STOC 2014).
Our result for testing Boolean functions over then follows by applying
the -query hypergrid
tester of Black-Chakrabarty-Seshadhri (SODA 2018).
To obtain the result for testing Boolean functions over , we
use standard measure theoretic tools to reduce monotonicity testing of a
measurable function to monotonicity testing of a discretized version of
over a hypergrid domain for large, but finite, (that may depend on
). The independence of in the hypergrid tester is crucial to getting the
final tester over
Fooling intersections of low-weight halfspaces
A weight- halfspace is a Boolean function sign where each is an integer in We give
an explicit pseudorandom generator that -fools any intersection of
weight- halfspaces with seed length poly. In
particular, our result gives an explicit PRG that fools any intersection of any
quasipoly number of halfspaces of any poly weight to any
poly accuracy using seed length poly Prior to this work
no explicit PRG with non-trivial seed length was known even for fooling
intersections of weight-1 halfspaces to constant accuracy.
The analysis of our PRG fuses techniques from two different lines of work on
unconditional pseudorandomness for different kinds of Boolean functions. We
extend the approach of Harsha, Klivans and Meka \cite{HKM12} for fooling
intersections of regular halfspaces, and combine this approach with results of
Bazzi \cite{Bazzi:07} and Razborov \cite{Razborov:09} on bounded independence
fooling CNF formulas. Our analysis introduces new coupling-based ingredients
into the standard Lindeberg method for establishing quantitative central limit
theorems and associated pseudorandomness results.Comment: 27 page
Free-Boolean independence for pairs of algebras
We construct pairs of algebras with mixed independence relations by using
truncations of reduced free products of algebras. For example, we construct
free-Boolean pairs of algebras and free-monotone pairs of algebras. We also
introduce free-Boolean cumulants and show that free-Boolean independence is
equivalent to the vanishing of mixed cumulants.Comment: Moments-condition for free-Boolean independence is added to Section
4. Title is changed. All comments are welcom
Reduction of Database Independence to Dividing in Atomless Boolean Algebras
We prove that the form of conditional independence at play in database theory
and independence logic is reducible to the first-order dividing calculus in the
theory of atomless Boolean algebras. This establishes interesting connections
between independence in database theory and stochastic independence. As indeed,
in light of the aforementioned reduction and recent work of Ben-Yaacov [4], the
former case of independence can be seen as the discrete version of the latter
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