Showing papers on "Boolean function published in 2016"

The Power of Depth for Feedforward Neural Networks

Abstract: We show that there is a simple (approximately radial) function on R d , expressible by a small 3-layer feedforward neural networks, which cannot be approximated by any 2-layer network, to more than a certain constant accuracy, unless its width is exponential in the dimension. The result holds for virtually all known activation functions, including rectified linear units, sigmoids and thresholds, and formally demonstrates that depth ‐ even if increased by 1 ‐ can be exponentially more valuable than width for standard feedforward neural networks. Moreover, compared to related results in the context of Boolean functions, our result requires fewer assumptions, and the proof techniques and construction are very different.

Bitwise Neural Networks

Abstract: Based on the assumption that there exists a neural network that efficiently represents a set of Boolean functions between all binary inputs and outputs, we propose a process for developing and deploying neural networks whose weight parameters, bias terms, input, and intermediate hidden layer output signals, are all binary-valued, and require only basic bit logic for the feedforward pass. The proposed Bitwise Neural Network (BNN) is especially suitable for resource-constrained environments, since it replaces either floating or fixed-point arithmetic with significantly more efficient bitwise operations. Hence, the BNN requires for less spatial complexity, less memory bandwidth, and less power consumption in hardware. In order to design such networks, we propose to add a few training schemes, such as weight compression and noisy backpropagation, which result in a bitwise network that performs almost as well as its corresponding real-valued network. We test the proposed network on the MNIST dataset, represented using binary features, and show that BNNs result in competitive performance while offering dramatic computational savings.

Bent Functions: Fundamentals and Results

Abstract: This book gives a detailed survey of the main results on bent functions over finite fields, presents a systematic overview of their generalizations, variations and applications, considers open problems in classification and systematization of bent functions, and discusses proofs of several results. This book uniquely provides a necessary comprehensive coverage of bent functions. It serves as a useful reference for researchers in discrete mathematics, coding and cryptography. Students and professors in mathematics and computer science will also find the content valuable, especially those interested in mathematical foundations of cryptography. It can be used as a supplementary text for university courses on discrete mathematics, Boolean functions, or cryptography, and is appropriate for both basic classes for under-graduate students and advanced courses for specialists in cryptography and mathematics.

Separations in query complexity using cheat sheets

Abstract: We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique, which builds upon the techniques of Ambainis et al. [STOC 2016]. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al. [STOC 2016], to be shown in a unified manner.

Stochastic p-bits for Invertible Logic

Abstract: Conventional logic and memory devices are built out of deterministic units such as transistors, or magnets with energy barriers in excess of 40-60 kT. We show that stochastic units, p-bits, can be interconnected to create robust correlations that implement Boolean functions with impressive accuracy, comparable to standard circuits. Also they are invertible, a unique property that is absent in digital circuits. When operated in the direct mode, the input is clamped, and the network provides the correct output. In the inverted mode, the output is clamped, and the network fluctuates among possible inputs consistent with that output. We present an implementation of an invertible gate to bring out the key role of a three-terminal building block to enable the construction of correlated p-bit networks. The results for this implementation agree well with those from a universal model, showing that p-bits need not be magnet-based: any three-terminal tunable random bit generator should be suitable. We present an algorithm for designing a Boltzmann machine (BM) with symmetric connections that implements a given truth table. We then show how BM Full Adders can be interconnected in a partially directed manner to implement large operations such as 32-bit addition. Hundreds of p-bits get precisely correlated such that the correct answer out of 2^33 possibilities can be extracted by looking at the mode of a number of time samples. With perfect directivity a small number of samples is enough, while for less directed connections more samples are needed, but even in the former case invertibility is largely preserved. This combination of accuracy and invertibility is enabled by the hybrid design that uses bidirectional units to construct circuits with partially directed connections. We establish this result with examples including a 4-bit multiplier which in inverted mode functions as a factorizer.

Shuffles and Circuits: (On Lower Bounds for Modern Parallel Computation)

Abstract: The goal of this paper is to identify fundamental limitations on how efficiently algorithms implemented on platforms such as MapReduce and Hadoop can compute the central problems in the motivating application domains, such as graph connectivity problems.We introduce an abstract model of massively parallel computation, where essentially the only restrictions are that the "fan-in" of each machine is limited to s bits, where s is smaller than the input size n, and that computation proceeds in synchronized rounds, with no communication between different machines within a round. Lower bounds on the round complexity of a problem in this model apply to every computing platform that shares the most basic design principles of MapReduce-type systems.We prove that computations in our model that use few rounds can be represented as low-degree polynomials over the reals. This connection allows us to translate a lower bound on the (approximate) polynomial degree of a Boolean function to a lower bound on the round complexity of every (randomized) massively parallel computation of that function. These lower bounds apply even in the "unbounded width" version of our model, where the number of machines can be arbitrarily large. As one example of our general results, computing any non-trivial monotone graph property --- such as connectivity --- requires a super-constant number of rounds when every machine can accept only a sub-polynomial (in n) number of input bits s.Finally, we prove that, in two senses, our lower bounds are the best one could hope for. For the unbounded-width model, we prove a matching upper bound. Restricting to a polynomial number of machines, we show that asymptotically better lower bounds require proving that P ≠ NC1.

Cryptographic Boolean functions

Abstract: Graphical abstractDisplay Omitted HighlightsAnalysis on the efficiency of EAs when evolving cryptographic Boolean functions.Three fitness functions, four evolutionary algorithms.Wide set of experiments with different algorithm parameter values.The best results are obtained with GP and CGP.We show the problem representation plays a crucial role for reaching top solutions. Boolean functions represent an important primitive in the design of various cryptographic algorithms. There exist several well-known schemes where a Boolean function is used to add nonlinearity to the cipher. Thus, methods to generate Boolean functions that possess good cryptographic properties present an important research goal. Among other techniques, evolutionary computation has proved to be a well-suited approach for this problem. In this paper, we present three different objective functions, where each inspects important cryptographic properties of Boolean functions, and examine four evolutionary algorithms. Our research confirms previous results, but also sheds new insights on the effectiveness and comparison of different evolutionary algorithms for this problem.

Rectangles Are Nonnegative Juntas

Abstract: We develop a new method to prove communication lower bounds for composed functions of the form $f\circ g^n$, where $f$ is any boolean function on $n$ inputs and $g$ is a sufficiently “hard” two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of $f \circ g^n$ can be simulated by a nonnegative combination of juntas. This is a new formalization for the intuition that each low-communication randomized protocol can only “query” a few inputs of $f$ as encoded by the gadget $g$. Consequently, we characterize the communication complexity of $f\circ g^n$ in all known one-sided (i.e., not closed under complement) zero-communication models by a corresponding query complexity measure of $f$. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work. We show that $\mathsf{SBP}^{\sf cc}$ (a class characterized...

An Orthogonal Basis for Functions over a Slice of the Boolean Hypercube

Abstract: We present a simple, explicit orthogonal basis of eigenvectors for the Johnson and Kneser graphs, based on Young's orthogonal representation of the symmetric group. Our basis can also be viewed as an orthogonal basis for the vector space of all functions over a slice of the Boolean hypercube (a set of the form $\{(x_1,\ldots,x_n) \in \{0,1\}^n : \sum_i x_i = k\}$), which refines the eigenspaces of the Johnson association scheme; our basis is orthogonal with respect to any exchangeable measure. More concretely, our basis is an orthogonal basis for all multilinear polynomials $\mathbb{R}^n \to \mathbb{R}$ which are annihilated by the differential operator $\sum_i \partial/\partial x_i$. As an application of the last point of view, we show how to lift low-degree functions from a slice to the entire Boolean hypercube while maintaining properties such as expectation, variance and $L^2$-norm. As an application of our basis, we streamline Wimmer's proof of Friedgut's theorem for the slice. Friedgut's theorem, a fundamental result in the analysis of Boolean functions, states that a Boolean function on the Boolean hypercube with low total influence can be approximated by a Boolean junta (a function depending on a small number of coordinates). Wimmer generalized this result to slices of the Boolean hypercube, working mostly over the symmetric group, and utilizing properties of Young's orthogonal representation. Using our basis, we show how the entire argument can be carried out directly on the slice.

How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity)

Abstract: We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constantsize coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large coefficients. These are also the first trade-offs to hold uniformly for resolution, polynomial calculus and cutting planes, thus capturing the main methods of reasoning used in current state-of-the-art SAT solvers. We prove our results by a reduction to communication lower bounds in a round-efficient version of the real communication model of [Kraj´iˇcek ’98], drawing on and extending techniques in [Raz and McKenzie ’99] and [G¨o¨os et al. ’15]. The communication lower bounds are in turn established by a reduction to trade-offs between cost and number of rounds in the game of [Dymond and Tompa ’85] played on directed acyclic graphs. As a by-product of the techniques developed to show these proof complexity trade-off results, we also obtain an exponential separation between monotone-ACi1 and monotone-ACi, improving exponentially over the superpolynomial separation in [Raz and McKenzie ’99]. That is, we give an explicit Boolean function that can be computed by monotone Boolean circuits of depth logi n and polynomial size, but for which circuits of depth O(logi1 n) require exponential size.

An efficient method for multi-level approximate logic synthesis under error rate constraint

Abstract: Approximate computing is an emerging design paradigm targeting at error-tolerant applications. It trades off accuracy for improvement in hardware cost and energy efficiency. In this paper, we propose a novel approach for multi-level approximate logic synthesis under error rate constraint. The basic idea of our approach is to pick nodes in a Boolean network and shrink them by approximating their factored-form expressions. We propose two different algorithms to implement the basic idea. The first algorithm iteratively picks the most effective node at present to shrink. Its drawback lies in that it may need a large number of iterations. To overcome this drawback, the second algorithm formulates a knapsack problem to pick multiple nodes for shrinking simultaneously. It is still iterative, but the number of iterations is greatly reduced. We apply the two algorithms to MCNC benchmarks and arithmetic circuits including adders and multipliers. The experimental results demonstrated that our algorithms perform better in area saving and are 1.7 and 5.9 times faster, respectively, compared with the state-of-the-art approach.

Separations in query complexity based on pointer functions

Abstract: In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n=2k bits defined by a complete binary tree of NAND gates of depth k, which achieves R0(f) = O(D(f)0.7537…). We show this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is Ω(n/log(n)) while its zero-error randomized query complexity is O(√n). We further show that the quantum query complexity of the same function is O(n1/4), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total boolean function g on n variables that has zero-error randomized query complexity Ω(n/log(n)) and bounded-error randomized query complexity R(g) = O(√n). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is QE(g) = O(√n). These functions show that the relations D(f) = O(R1(f)2) and R0(f) = O(R(f)2) are optimal, up to poly-logarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R0, a 3/2-power separation between QE and R, and a 4th power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by Goos, Pitassi, and Watson which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

More constructions of differentially 4-uniform permutations on $${\mathbb {F}}_{2^{2k}}$$F22k

Abstract: Differentially $$4$$4-uniform permutations on $${\mathbb {F}}_{2^{2k}}$$F22k with high nonlinearity are chosen as Substitution boxes in many block ciphers and some stream ciphers. Recently, Qu et al. (IEEE Trans Inf Theory, 59(7), 4675---4686, 2013) introduced a class of functions, which are called preferred functions, to construct a lot of infinite families of such permutations. In this paper, we propose a particular type of Boolean functions to characterize the preferred functions. On the one hand, such Boolean functions can be determined by solving linear equations, and they give rise to a huge number of differentially $$4$$4-uniform permutations over $${\mathbb {F}}_{2^{2k}}$$F22k. Hence they may provide more choices for the design of Substitution boxes. On the other hand, by investigating the number of these Boolean functions, we show that the number of CCZ-inequivalent differentially $$4$$4-uniform permutations over $${\mathbb {F}}_{2^{2k}}$$F22k grows exponentially when $$k$$k increases, which gives a positive answer to an open problem proposed in Qu et al.(IEEE Trans Inf Theory, 59(7), 4675---4686, 2013).

Relative stability of network states in Boolean network models of gene regulation in development.

Abstract: Progress in cell type reprogramming has revived the interest in Waddington's concept of the epigenetic landscape. Recently researchers developed the quasi-potential theory to represent the Waddington's landscape. The Quasi-potential U(x), derived from interactions in the gene regulatory network (GRN) of a cell, quantifies the relative stability of network states, which determine the effort required for state transitions in a multi-stable dynamical system. However, quasi-potential landscapes, originally developed for continuous systems, are not suitable for discrete-valued networks which are important tools to study complex systems. In this paper, we provide a framework to quantify the landscape for discrete Boolean networks (BNs). We apply our framework to study pancreas cell differentiation where an ensemble of BN models is considered based on the structure of a minimal GRN for pancreas development. We impose biologically motivated structural constraints (corresponding to specific type of Boolean functions) and dynamical constraints (corresponding to stable attractor states) to limit the space of BN models for pancreas development. In addition, we enforce a novel functional constraint corresponding to the relative ordering of attractor states in BN models to restrict the space of BN models to the biological relevant class. We find that BNs with canalyzing/sign-compatible Boolean functions best capture the dynamics of pancreas cell differentiation. This framework can also determine the genes' influence on cell state transitions, and thus can facilitate the rational design of cell reprogramming protocols.

BDD minimization for approximate computing

Abstract: We present Approximate BDD Minimization (ABM) as a problem that has application in approximate computing. Given a BDD representation of a multi-output Boolean function, ABM asks whether there exists another function that has a smaller BDD representation but meets a threshold w.r.t. an error metric. We present operators to derive approximated functions and present algorithms to exactly compute the error metrics directly on the BDD representation. An experimental evaluation demonstrates the applicability of the proposed approaches.

Boolean Functions and Their Applications in Cryptography

Abstract: This book focuses on the different representations and cryptographic properties of Booleans functions, presents constructions of Boolean functions with some good cryptographic properties. More specifically, Walsh spectrum description of the traditional cryptographic properties of Boolean functions, including linear structure, propagation criterion, nonlinearity, and correlation immunity are presented. Constructions of symmetric Boolean functions and of Boolean permutations with good cryptographic properties are specifically studied. This book is not meant to be comprehensive, but with its own focus on some original research of the authors in the past. To be self content, some basic concepts and properties are introduced. This book can serve as a reference for cryptographic algorithm designers, particularly the designers of stream ciphers and of block ciphers, and for academics with interest in the cryptographic properties of Boolean functions.

BDD-Based Boolean Functional Synthesis

Abstract: Boolean functional synthesis is the process of automatically obtaining a constructive formalization from a declarative relation that is given as a Boolean formula. Recently, a framework was proposed for Boolean functional synthesis that is based on Craig Interpolation and in which Boolean functions are represented as And-Inverter Graphs (AIGs). In this work we adapt this framework to the setting of Binary Decision Diagrams (BDDs), a standard data structure for representation of Boolean functions. Our motivation in studying BDDs is their common usage in temporal synthesis, a fundamental technique for constructing control software/hardware from temporal specifications, in which Boolean synthesis is a basic step. Rather than using Craig Interpolation, our method relies on a technique called Self-Substitution, which can be easily implemented by using existing BDD operations. We also show that this yields a novel way to perform quantifier elimination for BDDs. In addition, we look at certain BDD structures called input-first, and propose a technique called TrimSubstitute, tailored specifically for such structures. Experiments on scalable benchmarks show that both Self-Substitution and TrimSubstitute scale well for benchmarks with good variable orders and significantly outperform current Boolean-synthesis techniques.

An $o(n)$ Monotonicity Tester for Boolean Functions over the Hypercube

Abstract: A Boolean function $f:\{0,1\}^n \mapsto \{0,1\}$ is said to be $\varepsilon$-far from monotone if $f$ needs to be modified in at least $\varepsilon$-fraction of the points to make it monotone. We design a randomized tester that is given oracle access to $f$ and an input parameter $\varepsilon>0$ and has the following guarantee: It outputs \sf Yes if the function is monotonically nondecreasing and outputs \sf No with probability $>2/3$, if the function is $\varepsilon$-far from monotone. This nonadaptive, one-sided tester makes $O(n^{7/8}\varepsilon^{-3/2}\ln(1/\varepsilon))$ queries to the oracle.

Evolutionary algorithms for boolean functions in diverse domains of cryptography

Abstract: The role of Boolean functions is prominent in several areas including cryptography, sequences, and coding theory. Therefore, various methods for the construction of Boolean functions with desired properties are of direct interest. New motivations on the role of Boolean functions in cryptography with attendant new properties have emerged over the years. There are still many combinations of design criteria left unexplored and in this matter evolutionary computation can play a distinct role. This article concentrates on two scenarios for the use of Boolean functions in cryptography. The first uses Boolean functions as the source of the nonlinearity in filter and combiner generators. Although relatively well explored using evolutionary algorithms, it still presents an interesting goal in terms of the practical sizes of Boolean functions. The second scenario appeared rather recently where the objective is to find Boolean functions that have various orders of the correlation immunity and minimal Hamming weight. In both these scenarios we see that evolutionary algorithms are able to find high-quality solutions where genetic programming performs the best.

Very narrow quantum OBDDs and width hierarchies for classical OBDDs

Abstract: In the paper we investigate a model for computing of Boolean functions – Ordered Binary Decision Diagrams (OBDDs), which is a restricted version of Branching Programs. We present several results on the comparative complexity for several variants of OBDD models. We present some results on the comparative complexity of classical and quantum OBDDs. We consider a partial function depending on a parameter k such that for any k > 0 this function is computed by an exact quantum OBDD of width 2, but any classical OBDD (deterministic or stable bounded-error probabilistic) needs width 2 k + 1. We consider quantum and classical nondeterminism. We show that quantum nondeterminism can be more efficient than classical nondeterminism. In particular, an explicit function is presented which is computed by a quantum nondeterministic OBDD with constant width, but any classical nondeterministic OBDD for this function needs non-constant width. We also present new hierarchies on widths of deterministic and nondeterministic OBDDs. We focus both on small and large widths.

Driven Stability of Nonlinear Feedback Shift Registers With Inputs

Abstract: Driven stable nonlinear feedback shift registers (NFSRs) with inputs are not only able to limit error propagations in convolutional decoders, but also helpful to analyze the period properties of sequences generated by a cascade connection of NFSRs in stream ciphers. An NFSR is driven stable if and only if the reachable set is a subset of the basin. Due to lack of efficient algebraic tools, the driven stability of NFSRs with inputs has been much less studied. This paper continues to address this research using a Boolean control network approach. Viewing an NFSR with input as a Boolean control network, we first give its Boolean control network representation, which is characterized with a state transition matrix. Some properties of the state transition matrix are then provided. Based on these, explicit forms are given for the reachable set and the set of basin. Two algorithms for obtaining both the sets are provided as well. Compared with the exhaustive search and the existing state operator method, the Boolean control network approach requires lower computational complexity for those NFSRs with their stages greater than 1.

Approximation Algorithms for Stochastic Submodular Set Cover with Applications to Boolean Function Evaluation and Min-Knapsack

Abstract: We present a new approximation algorithm for the stochastic submodular set cover (SSSC) problem called adaptive dual greedy. We use this algorithm to obtain a 3-approximation algorithm solving the stochastic Boolean function evaluation (SBFE) problem for linear threshold formulas (LTFs). We also obtain a 3-approximation algorithm for the closely related stochastic min-knapsack problem and a 2-approximation for a variant of that problem.We prove a new approximation bound for a previous algorithm for the SSSC problem, the adaptive greedy algorithm of Golovin and Krause.We also consider an approach to approximating SBFE problems using the adaptive greedy algorithm, which we call the Q-value approach. This approach easily yields a new result for evaluation of CDNF (conjunctive / disjunctive normal form) formulas, and we apply variants of it to simultaneous evaluation problems and a ranking problem. However, we show that the Q-value approach provably cannot be used to obtain a sublinear approximation factor for the SBFE problem for LTFs or read-once disjunctive normal form formulas.

Stability of nonlinear feedback shift registers

Abstract: Convolutional codes have been widely used in many applications such as digital video, radio, and mobile communication. Nonlinear feedback shift registers (NFSRs) are the main building blocks in convolutional decoders. A decoding error may result in a succession of further decoding errors. However, a stable NFSR can limit such an error-propagation. This paper studies the stability of NFSRs using a Boolean network approach. A Boolean network is an autonomous system that evolves as an automaton through Boolean functions. An NFSR can be viewed as a Boolean network. Based on its Boolean network representation, some sufficient and necessary conditions are provided for globally (locally) stable NFSRs. To determine the global stability of an NFSR with its stage greater than 1, the Boolean network approach requires lower time complexity of computations than the exhaustive search and the Lyapunov’s direct method.

The Complexity of DNF of Parities

Abstract: We study depth 3 circuits of the form OR-AND-XOR, or equivalently -- DNF of parities. This model was first explicitly studied by Jukna (CPC'06) who obtained a 2{Ω(n) lower bound, using graph theoretic arguments, for explicit functions. Several related models have gained attention in the last few years, such as parity decision trees, the parity kill number and AC0-XOR circuits.For a Boolean function f on the n dimensional Boolean cube, we denote by DNFParity(f) the least integer s for which there exists an OR-AND-XOR circuit, with OR gate of fan-in s, that computes f. We summarize some of our results: For any affine disperser f for dimension k, it holds that DNFParity(f) is bounded below by 2{n-2k. By plugging Shaltiel's affine disperser (FOCS'11) we obtain an explicit 2{n-no(1) lower bound.We give a non-trivial general upper bound by showing that DNFParity(f)

A quantum annealing approach for boolean satisfiability problem

Abstract: Quantum annealing device has shown a great potential in solving discrete problems that are theoretically and empirically hard. Boolean Satisfiability (SAT) problem, determining if there is an assignment of variables that satisfies a given Boolean function, is the first proven NP-complete problem widely used in various domains. Here, we present a novel mapping of the SAT problem to the quadratic unconstrained binary optimization problem (QUBO), and further develop a tool flow embedding the proposed QUBO to the architecture of the commercialized quantum computer D-Wave. By leveraging electronic design automation techniques including synthesis, placement and routing, this is not only the first work providing the detail flow that embeds the QUBO, but also a technique scalable for real world applications and some hard SAT problems with over 6000 variables in QUBO. Based on our results, we discuss the challenges in solving SAT using the current generation of annealing device, and explore the problem solving capability of future quantum annealing computers.

Learning sparse two-level boolean rules

Abstract: This paper develops a novel optimization framework for learning accurate and sparse two-level Boolean rules for classification, both in Conjunctive Normal Form (CNF, i.e. AND-of-ORs) and in Disjunctive Normal Form (DNF, i.e. OR-of-ANDs). In contrast to opaque models (e.g. neural networks), sparse two-level Boolean rules gain the crucial benefit of interpretability, which is necessary in a wide range of applications such as law and medicine and is attracting considerable attention in machine learning. This paper introduces two principled objective functions to trade off classification accuracy and sparsity, where 0-1 error and Hamming loss are used to characterize accuracy. We propose efficient procedures to optimize these objectives based on linear programming (LP) relaxation, block coordinate descent, and alternating minimization. We also describe a new approach to rounding any fractional values in the optimal solutions of LP relaxations. Experiments show that our new algorithms based on the Hamming loss objective provide excellent tradeoffs between accuracy and sparsity with improvements over state-of-the-art methods.