Related papers: Nearest Neighbor Complexity and Boolean Circuits

Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in…

Boolean circuit is a computational graph that consists of the dynamic directed graph structure and static functionality. The commonly used logic optimization and Boolean matching-based transformation can change the behavior of the Boolean…

We consider the problem of testing whether a Boolean function has Fourier degree $\leq k$ or it is $\epsilon$-far from any Boolean function with Fourier degree $\leq k$. We improve the known lower bound of $\Omega(k)$ \cite{BBM11,CGM10}, to…

The sensitivity of a Boolean function f is the maximum over all inputs x, of the number of sensitive coordinates of x. The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-s Boolean…

In this work, we consider a new type of Fourier-like representation of Boolean function $f\colon\{+1,-1\}^n\to\{+1,-1\}$ \[ f(x) = \cos\left(\pi\sum_{S\subseteq[n]}\phi_S \prod_{i\in S} x_i\right). \] This representation, which we call the…

We study the power of negation in the Boolean and algebraic settings and show the following results. * We construct a family of polynomials $P_n$ in $n$ variables, all of whose monomials have positive coefficients, such that $P_n$ can be…

We establish a lower bound for deciding the satisfiability of the conjunction of any two Boolean formulas from a set called a full representation of Boolean functions of $n$ variables - a set containing a Boolean formula to represent each…

Probabilistic k-nearest neighbour (PKNN) classification has been introduced to improve the performance of original k-nearest neighbour (KNN) classification algorithm by explicitly modelling uncertainty in the classification of each feature…

This paper addresses the problem of finding a B-term wavelet representation of a given discrete function $f \in \real^n$ whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance…

We study the problem of computing the minimum adversarial perturbation of the Nearest Neighbor (NN) classifiers. Previous attempts either conduct attacks on continuous approximations of NN models or search for the perturbation by some…

The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial $\mathbf{P}$ of degree at most $d$ that agrees with $f$ at each point with high…

Nearest neighbor (kNN) methods have been gaining popularity in recent years in light of advances in hardware and efficiency of algorithms. There is a plethora of methods to choose from today, each with their own advantages and…

We show that sharp thresholds for Boolean functions directly imply average-case circuit lower bounds. More formally we show that any Boolean function exhibiting a sharp enough threshold at \emph{arbitrary} critical density cannot be…

Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ gates (for some constant $m > 1$), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen…

Demands for minimum parameter setup in machine learning models are desirable to avoid time-consuming optimization processes. The $k$-Nearest Neighbors is one of the most effective and straightforward models employed in numerous problems.…

Suppose $V$ is an $n$-element set where for each $x \in V$, the elements of $V \setminus \{x\}$ are ranked by their similarity to $x$. The $K$-nearest neighbor graph is a directed graph including an arc from each $x$ to the $K$ points of $V…

We prove two main results on how arbitrary linear threshold functions $f(x) = \sign(w\cdot x - \theta)$ over the $n$-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every…

The minimum number of NOT gates in a Boolean circuit computing a Boolean function is called the inversion complexity of the function. In 1957, A. A. Markov determined the inversion complexity of every Boolean function and proved that…

In this paper, we establish a new bound tying together the effective length and the maximum correlation between the outputs of an arbitrary pair of Boolean functions which operate on two sequences of correlated random variables. We derive a…

The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov recently characterized the minimal degree deg_{\eps}(f) among all polynomials (over the…