On the constructive characterization of threshold functions

Unlike inflexible structure of soft and hard threshold function, a unified linear matrix form with flexible structure for threshold function is proposed. Based on the unified linear flexible structure threshold function, both supervised and unsupervised subband adaptive denoising frameworks are established. To determine flexible coefficients, a direct mean-square error (MSE) minimization is conducted in supervised denoising while Stein’s unbiased risk estimate as a MSE estimate is minimized in unsupervised denoising. The SURE rule requires no hypotheses or a priori knowledge about clean signals. Furthermore, we discuss conditions to obtain optimal coefficients for both supervised and unsupervised subband adaptive denoising frameworks. Applying an Odd-Term Reserving Polynomial (OTRP) function as concrete threshold function, simulations for polynomial order, denoising performance, and noise effect are conducted. Proper polynomial order and noise effect are analyzed. Both proposed methods are compared with soft and hard based denoising technologies—VisuShrink, SureShrink, MiniMaxShrink, and BayesShrink—in denoising performance simulation. Results show that the proposed approaches perform better in both MSE and signal-to-noise ratio (SNR) sense.

Dimension reduction and classification of data is an important issue in the data processing. Reduct in the rough set is a minimal subset of features, which has the same discernible power as the entire features in the higher dimensional scheme. Boolean threshold function was of fundamental interest for classification function. The nearest neighbor relation with minimal distance proposed here has a fundamental information for classification. Further, the nearest neighbor relation plays a crucial role for generation of reducts and threshold functions using the Boolean reasoning on the discernibility and in discernibility matrices, in which the indiscernibility matrix is proposed here to test the sufficient condition for reduct and threshold function. Then, a genetation method for the threshold functions based on the nearest neighbor relation are proposed here using Boolean operations.

1 - Linear Algebra

Many voting systems consist of a set of players who can form coalitions; a winning coalition is one that can pass a measure. The winning and losing coalitions can be given by several classes of functions, including switching functions, threshold functions, weighted voting systems, and simple games. We define a zonotope T n T_n related to these functions and describe its coordinates in terms of a measure of voting power. We also give a voting interpretation of the coordinates of the derived zonotope T ¯ n \bar {T}_n .

Recent research has investigated time-space tradeoffs for register allocation strategies of certain fixed sets of expressions. This paper is concerned with the time-space tradeoff for register allocation strategies of any set of expressions which compute given functions. Time-space tradeoffs for pebbling superconcentrators and grates are developed. Corollaries which follow include tradeoffs for any straight-line program which computes polynomial multiplication, polynomial convolution, the discrete Fourier transform, oblivious merging, and most sets of linear forms.

We show that the rank of a depth-$3$ circuit (over any field) that is simple, minimal and zero is at most $O(k^3\log d)$. The previous best rank bound known was $2^{O(k^2)}(\log d)^{k-2}$ by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank $\Omega(k\log d)$). Our rank bound significantly improves (dependence on $k$ exponentially reduced) the best known deterministic black-box identity tests for depth-$3$ circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth-$3$ circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth-$3$ circuit (over any field) is at most $O(k^3\log d)$. The novel feature of this work is a new notion of maps between sets of linear forms, called \emph{ideal matchings}, used to study depth-$3$ circuits. We prove interesting structural results about depth-$3$ identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits.

The non-Lefschetz locus of vector bundles of rank 2 over [formula omitted

Boundary points of a threshold function fare those vertices of the n-cube that produce a minimal irredundant set of inequalities for the realization of f. In this paper they are shown to be those vertices that can be contained in separating hyperplanes. With this theorem it is shown that knowledge of canonical boundary points of self-dual canonical threshold functions allows complete determination of boundary points of equivalent functions. This provides a compact and geometrically interesting characterization of threshold functions.

We study the complexity of computing Boolean functions using AND, OR and NOT gates. We show that a circuit of depth d with S gates can be made to output a constant by setting O(S1− ) (where (d) = 4−d) of its input values. This implies a superlinear size lower bound for a large class of functions. Using this, we obtain a function computable by a uniform family of constant depth polynomial size circuits that cannot be computed by constant depth circuits of linear size. We give circuit constructions that show that the bound O(S1− ) is near optimal. We also study the complexity of computing threshold functions. The function T r has the value 1 iff at least r of its inputs have the value 1. We show that a circuit computing T r has at least Ω(r (logn)/ log r) gates, for r ≤ n, improving previous bounds. We also show a trade-off between the number of gates and the number of wires in a threshold circuit, namely, a circuit with G (< n/2) gates and W wires computing T r satisfies W ≥ Ω(nr(logn)/(log(G/ logn))), showing that it is not possible to simultaneously optimize the number of gates and wires in a threshold circuit. Our bounds for threshold functions are based on a combinatorial lemma of independent interest.

The Conversion Set Problem on Graphs

It is well-known that deciding consistency for normal answer set programs (ASP) is NP-complete, thus, as hard as the satisfaction problem for propositional logic (SAT). The exponential time hypothesis (ETH) implies that the best algorithms to solve these problems take exponential time in the worst case. However, accounting for the treewidth, the consistency problem for ASP is slightly harder than SAT: while SAT can be solved by an algorithm that runs in exponential time in the treewidth k, ASP requires exponential time in k · log(k). This extra cost is due to checking that there are no self-supported true atoms due to positive cycles in the program. In this paper, we refine this recent result and show that consistency for ASP can be decided in exponential time in k · log(ι) where ι is a novel measure, bounded by both treewidth k and the size of the largest strongly-connected component of the positive dependency graph of the program. We provide a treewidth-aware reduction from ASP to SAT that adheres to the above limit.

Phase-field theories for mathematical modeling of biological membranes

Here is a direct problem for Sidon sets: Given a linear form $$\varphi = c_1 x_1 + \cdots + c_h x_h $$ , construct and describe sets A that are Sidon sets for $$\varphi $$ . This paper considers an inverse problem for Sidon sets: Given a set A, determine the linear forms $$\varphi $$ such that A is a Sidon set for $$\varphi $$ .

CHAPTER 2 - A Catalog of Complexity Classes

This paper has two main focal points. We first consider an important class of machine learning algorithms: large margin classifiers, such as Support Vector Machines. The notion of margin complexity quantifies the extent to which a given class of functions can be learned by large margin classifiers. We prove that up to a small multiplicative constant, margin complexity is equal to the inverse of discrepancy. This establishes a strong tie between seemingly very different notions from two distinct areas.In the same way that matrix rigidity is related to rank, we introduce the notion of rigidity of margin complexity. We prove that sign matrices with small margin complexity rigidity are very rare. This leads to the question of proving lower bounds on the rigidity of margin complexity. Quite surprisingly, this question turns out to be closely related to basic open problems in communication complexity, e.g., whether PSPACE can be separated from the polynomial hierarchy in communication complexity.Communication is a key ingredient in many types of learning. This explains the relations between the field of learning theory and that of communication complexity [6, l0, 16, 26]. The results of this paper constitute another link in this rich web of relations. These new results have already been applied toward the solution of several open problems in communication complexity [18, 20, 29].