Boolean Convolution Research Articles

Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product

Let F(νj)={Qmjνj,mj∈(m−νj,m+νj)}, j=1,2, be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures ν1 and ν2. Introduce the set of measures F=F(ν1)⊞F(ν2)={Qm1ν1⊞Qm2ν2,m1∈(m−ν1,m+ν1)andm2∈(m−ν2,m+ν2)}. We show that if F remains a CSK family, (i.e., F=F(μ) where μ is a non-degenerate compactly supported measure), then the measures μ, ν1 and ν2 are of the Marchenko–Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of F) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of F) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution are also studied.

Some Limiting Measures Related to Free and Boolean Convolutions

Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltjes Kernel (CSK) families of probabilities. By means of the variance functions machinery, some novel limiting probability measures are provided involving the free multiplicative law of large numbers, using both free and Boolean (additive and multiplicative) convolutions. Several examples of these limiting probability measures are given in the context of free probability.

3D-BCNN-Based Image Encryption With Finite Computing Precision

To resolve the finite computing precision problem in chaotic cryptography, and the security and efficiency drawbacks of deep learning-based image cryptosystems, an image encryption framework with low computing precision is presented based on three-dimensional Boolean convolution neural network (3D-BCNN). Unlike traditional CNN, the proposed 3D-BCNN is composed of only convolutional layers, in which a cross-channel 3-D Boolean convolution operation is devised without training and parameter optimization. To strengthen the security and sensitivity of the cryptosystem, the kernels and convolutional matrices are generated by the combined prime modulo multiplicative linear congruence generators, SHA-1, and a plain-image. All operations of 3D-BCNN can be conducted on devices with 8-bit word length, so the proposed encryption scheme could work well with low computing precision. Simulation results demonstrate that, with the largest computing accuracy 2−8, the proposed encryption scheme has high security and low time complexity, and can withstand various attacks.

A survey on the effects of free and boolean convolutions on Cauchy-Stieltjes Kernel families

In the setting of noncommutative probability theory and in analogy with the theory of natural exponential families (NEFs), a theory of Cauchy-Stieltjes Kernel (CSK) families has been recently introduced. It is based on the Cauchy-Stieltjes kernel (1−θx)−1. In this paper, after presenting some basic concepts on NEFs and CSK families and pointing out some similarities and differences between the two families, we review the present state and developments regarding the effects of free and boolean convolutions powers on CSK families.

Small normalized circuits for semi-disjoint bilinear forms require logarithmic and-depth

We consider normalized Boolean circuits that use binary operations of disjunction and conjunction, and unary negation, with the restriction that negation can be only applied to input variables. We derive a lower bound trade-off between the size of normalized Boolean circuits computing Boolean semi-disjoint bilinear forms and their conjunction-depth (i.e., the maximum number of and-gates on a directed path to an output gate). In particular, we show that any normalized Boolean circuit of at most ϵlog⁡n conjunction-depth computing the n-dimensional Boolean vector convolution has Ω(n2−4ϵ) and-gates. For Boolean matrix product, we derive even a stronger lower-bound trade-off. Instead of conjunction-depth we use the negation-dependent conjunction-depth, where one counts only and-gates whose each direct predecessor has a (not necessarily direct) predecessor representing a negated input variable. We show that if a normalized Boolean circuit of at most ϵlog⁡n negation-dependent conjunction-depth computes the n×n Boolean matrix product then the circuit has Ω(n3−2ϵ) and-gates. We complete our lower-bound trade-offs for the Boolean convolution and matrix product with upper-bound trade-offs of similar form yielded by the known fast algebraic algorithms.

Extreme Witnesses and Their Applications

We study the problem of computing the so called minimum and maximum witnesses for Boolean vector convolution. We also consider a generalization of the problem which is to determine for each positive value at a coordinate of the convolution vector, q smallest (largest) witnesses, where q is the minimum of a parameter k and the number of witnesses for this coordinate. We term this problem the smallest k-witness problem or the largest k-witness problem, respectively. We also study the corresponding smallest and largest k-witness problems for Boolean matrix product. First, we present an tilde{O}(n^{1.5}k^{0.5})-time algorithm for the smallest or largest k-witness problem for the Boolean convolution of two n-dimensional vectors, where the notation tilde{O}( ) suppresses polylogarithmic in n factors. In consequence, we obtain new upper time bounds on reporting positions of mismatches in potential string alignments and on computing restricted cases of the (min , +) vector convolution. Next, we present a fast (substantially subcubic in n and linear in k) algorithm for the smallest or largest k-witness problem for the Boolean matrix product of two ntimes n Boolean matrices. It yields fast algorithms for reporting k lightest (heaviest) triangles in a vertex-weighted graph.

Δ-Cumulants in terms of moments

The [Formula: see text]-convolution of real probability measures, introduced by Bożejko, generalizes both free and Boolean convolutions. It is linearized by the [Formula: see text]-cumulants, and Yoshida gave a combinatorial formula for moments in terms of [Formula: see text]-cumulants, that implicitly defines the latter. It relies on the definition of an appropriate weight on noncrossing partitions. We give here two different expressions for the [Formula: see text]-cumulants: the first one is a simple variant of Lagrange inversion formula, and the second one is a combinatorial inversion of Yoshida’s formula involving Schröder trees.

Boolean convolutions and regular variation

In this article we study the influence of regularly varying probability measures on additive and multiplicative Boolean convolutions. We introduce the notion of Boolean subexponentiality (for additive Boolean convolution), which extends the notion of classical and free subexponentiality. We show that the distributions with regularly varying tails belong to the class of Boolean subexponential distributions. As an application we also study the behaviour of the Belinschi-Nica map. Breiman's theorem study the classical product convolution between regularly varying measures. We derive an analogous result to Breiman's theorem in case of multiplicative Boolean convolution. In proving these results we exploit the relationship of regular variation with different transforms and their Taylor series expansion.

Boolean convolution in the quaternionic setting

In this paper we begin a study of free analysis in the quaternionic setting, and consider Boolean convolution for quaternion-valued measures. To this end we also study Boolean convolution for matrix-valued complex measures, also proving Boolean infinite divisibility and central limit theorems for these measures. Moreover we prove an integral representation for quaternionic Carathéodory and Herglotz functions.

Classical and free infinite divisibility for Boolean stable laws

We completely determine the free infinite divisibility for the Boolean stable law which is parametrized by a stability index $\alpha$ and an asymmetry coefficient $\rho$. We prove that the Boolean stable law is freely infinitely divisible if and only if one of the following conditions holds: $0<\alpha\leq\frac{1}{2}$; $\frac{1}{2}<\alpha\leq\frac{2}{3}$ and $2-\frac{1}{\alpha}\leq\rho \leq \frac{1}{\alpha}-1$; $\alpha=1,~\rho=\frac{1}{2}$. Positive Boolean stable laws corresponding to $\rho =1$ and $\alpha \leq \frac{1}{2}$ have completely monotonic densities and they are both freely and classically infinitely divisible. We also show that continuous Boolean convolutions of positive Boolean stable laws with different stability indices are also freely and classically infinitely divisible. Boolean stable laws, free stable laws and continuous Boolean convolutions of positive Boolean stable laws are non-trivial examples whose free divisibility indicators are infinity. We also find that the free multiplicative convolution of Boolean stable laws is again a Boolean stable law.

Semigroups related to additive and multiplicative, free and Boolean convolutions

Belinschi and Nica introduced a composition semigroup on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know whether a probability measure is freely infinitely divisible or not. In this paper we further investigate this indicator, introduce a multiplicative version of it and are able to show many properties. Specifically, on the first half of the paper, we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to infinity; we derive an upper bound of the indicator in terms of Jacobi parameters. This upper bound is achieved only by free Meixner distributions. We also prove Bozejko's conjecture which says the Boolean power of a probability measure mu by 0 < t < 1 is freely infinitely divisible if mu is so. In the other half of this paper, we introduce an analogous composition semigroup for multiplicative convolutions and define free divisibility indicators for these convolutions. Moreover, we prove that a probability measure on the unit circle is freely infinitely divisible concerning the multiplicative free convolution if and only if the indicator is not less than one. We also prove how the multiplicative divisibility indicator changes under free and Boolean powers and then the multiplicative analogue of Bozejko's conjecture. We include an appendix, where the Cauchy distributions and point measures are shown to be the only fixed points of the Boolean-to-free Bercovici-Pata bijection.

Convolution powers in the operator-valued framework

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with respect to Boolean convolution, where the exponent considered in the power is a suitably chosen linear map \eta from B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever \eta is completely positive, while the free additive convolution power is defined whenever \eta - 1 is completely positive (where 1 stands for the identity map on B). In connection to these convolution powers we define an evolution semigroup related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the B-valued free Brownian motion. We also obtain two results on the operator-valued analytic function theory related to the free additive convolution powers with exponent \eta. One of the results concerns analytic subordination for B-valued Cauchy-Stieltjes transforms. The other gives a B-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued free Brownian motion.

Conditionally Monotone Independence II: Multiplicative Convolutions and Infinite Divisibility

We study the multiplicative convolution for c-monotone independence. This convolution unifies the monotone, Boolean and orthogonal multiplicative convolutions. We characterize convolution semigroups for the c-monotone multiplicative convolution on the unit circle. We also prove that an infinitely divisible distribution can always be embedded in a convolution semigroup. We furthermore discuss the (non)-uniqueness of such embeddings including the monotone case. Finally connections to the multiplicative Boolean convolution are discussed.

Symmetrization of probability measures, pushforward of order 2 and the Boolean convolution

We study relations between the Boolean convolution and the symmetrization and the pushforward of order 2. In particular we prove that if $\mu_1,\mu_2$ are probability measures on $[0,\infty)$ then $\left(\mu_1\uplus\mu_2\right)^{\mathbf{s}} =\mu_1^{\mathb

On the non-classical infinite divisibility of power semicircle distributions

The family of power semicircle distributions defined as normal- ized real powers of the semicircle density is considered. The marginals of uniform distributions on spheres in high-dimensional Euclidean spaces are included in this family and a boundary case is the classical Gaussian dis- tribution. A review of some results including a genesis and the so-called Poincare's theorem is presented. The moments of these distributions are re- lated to the super Catalan numbers and their Cauchy transforms in terms of hypergeometric functions are derived. Some members of this class of dis- tributions play the role of the Gaussian distribution with respect to additive convolutions in non-commutative probability, such as the free, the monotone, the anti-monotone and the Boolean convolutions. The infinite divisibility of other power semicircle distributions with respect to these convolutions is stud- ied using simple kurtosis arguments. A connection between kurtosis and the free divisibility indicator is found. It is shown that for the classical Gaussian distribution the free divisibility indicator is strictly less than 2.

FREE BROWNIAN MOTION AND EVOLUTION TOWARDS ⊞-INFINITE DIVISIBILITY FOR k-TUPLES

Let [Formula: see text] be the space of non-commutative distributions of k-tuples of self-adjoint elements in a C*-probability space. For every t ≥ 0 we consider the transformation [Formula: see text] defined by [Formula: see text] where ⊞ and ⊎ are the operations of free additive convolution and respectively of Boolean convolution on [Formula: see text]. We prove that 𝔹s◦ 𝔹t= 𝔹s + t, for all s, t ≥ 0. For t = 1, we prove that [Formula: see text] is precisely the set [Formula: see text] of distributions in [Formula: see text] which are infinitely divisible with respect to ⊞, and that the map [Formula: see text] coincides with the multi-variable Boolean Bercovici–Pata bijection put into evidence in our previous paper [1]. Thus for a fixed [Formula: see text], the process {𝔹t(μ)|t ≥ 0} can be viewed as some kind of "evolution towards ⊞-infinite divisibility".On the other hand, we put into evidence a relation between the transformations ⊞tand free Brownian motion. More precisely, we introduce a map [Formula: see text] which transforms the free Brownian motion started at an arbitrary [Formula: see text] into the process {𝔹t(μ)|t ≥ 0} for μ = Φ(ν).

Monotone and boolean convolutions for non-compactly supported probability measures

The equivalence of the characteristic function approach and the probabilistic approach to monotone and boolean convolutions is proven for non-compactly supported probability measures. A probabilistically motivated definition of the multiplicative boolean convolution of probability measures on the positive half-line is proposed. Unlike Bercovici's multiplicative boolean convolution it is always defined, but it turns out to be neither commutative nor associative. Finally some relations between free, monotone, and boolean convolutions are discussed.

Limit laws for Boolean convolutions

We study the distributional behavior for products, and for sums of boolean independent random variables in an infinitesimal triangular array. We show that the limit laws of boolean convolutions are determined by the limit laws of free convolutions, and vice versa. We further use these results to show several connections between the limiting distributional behavior of classical convolutions and that of boolean convolutions. The proof of our results is based on the analytical apparatus developed for free convolutions.

Faster pattern matching with character classes using prime number encoding

In pattern matching with character classes the goal is to find all occurrences of a pattern of length m in a text of length n, where each pattern position consists of an allowed set of characters from a finite alphabet Σ. We present an FFT-based algorithm that uses a novel prime-numbers encoding scheme, which is log n / log m times faster than the fastest extant approaches, which are based on boolean convolutions. In particular, if m | Σ | = n O ( 1 ) , our algorithm runs in time O ( n log m ) , matching the complexity of the fastest techniques for wildcard matching, a special case of our problem. A major advantage of our algorithm is that it allows a tradeoff between the running time and the RAM word size. Our algorithm also speeds up solutions to approximate matching with character classes problems—namely, matching with k mismatches and Hamming distance, as well as to the subset matching problem.

On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

r. Let M denote the space of Borel probability measures on R. For every t > 0 we consider the transformation B t : M → W defined by B t (μ) =(μ□ (1+t) ) U(1/1+t) , μ∈Μ, where □ and U are the operations of free additive convolution and respectively of Boolean convolution on M, and where the convolution powers with respect to □ and w are defined in the natural way. We show that B s o B t = B s+t, ∀ s, t ≥ 0 and that, quite surprisingly, every B t is a homomorphism for the operation of free multiplicative convolution □ (that is, B t (μ 14 ν) = B t (μ) □ B t (ν) for all p, ν ∈ M such that at least one of μ, ν is supported on [0, oo)). We prove that for t = 1 the transformation B 1 coincides with the canonical bijection B: M → M inf-div discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here M inf-div stands for the set of probability distributions in M which are infinitely divisible with respect to the operation □. As a consequence, we have that B t (μ) is □-infinitely divisible for every μ ∈ M and every t ≥ 1. On the other hand we put into evidence a relation between the transformations B t and the free Brownian motion; indeed, Theorem 1.6 of the paper gives an interpretation of the transformations B t as a way of recasting the free Brownian motion, where the resulting process becomes multiplicative with respect to □, and always reaches □-infinite divisibility by the time t = 1.