max norm optimization for robust matrix recovery this paper studies the matrix completion problem under arbitrary sampling schemes we propose a new estimator incorporating both max norm and nuclear norm regularization based on which we can conduct efficient low rank matrix recovery using a random subset of entries observed with additive noise under general non uniform and unknown sampling distributions this method significantly relaxes the uniform sampling assumption imposed for the widely used nuclear norm penalized approach and makes low rank matrix recovery feasible in more practical settings theoretically we prove that the proposed estimator achieves fast rates of convergence under different settings computationally we propose an alternating direction method of multipliers algorithm to efficiently compute the estimator which bridges a gap between theory and practice of machine learning methods with max norm regularization further we provide thorough numerical studies to evaluate the proposed method using both simulated and real datasets

an interpolating memory unit for function evaluation analysis and design a technique for the evaluation of a general continuous function f x is presented and the design of an interpolating memory as an implementation of the technique is described the technique partitions the domain of f x into segments and defines an interpolating or approximating function for each the implementation is a memory subsystem that holds the parameters of the approximating functions and yields an interpolated function value on each read reference polynomial interpolating functions are considered in particular hardware requirements memory and computational logic are analyzed in terms of the required precision it is shown that as long as f x has d 1 derivatives where d is the degree of the interpolating polynomial d 1 additional bits of precision of the computed f x are obtained for each additional address bit used in the interpolating memory this establishes a tradeoff between memory and computational logic which can be exploited in the design of a unit for a specific function for any precision requirement furthermore a single unit can be designed for any class of functions that have the required derivatives two examples of implementations for particular functions are presented

using underapproximations for sparse nonnegative matrix factorization nonnegative matrix factorization consists in approximately factorizing a nonnegative data matrix by the product of two low rank nonnegative matrices it has been successfully applied as a data analysis technique in numerous domains e g text mining image processing microarray data analysis collaborative filtering etc we introduce a novel approach to solve nmf problems based on the use of an underapproximation technique and show its effectiveness to obtain sparse solutions this approach based on lagrangian relaxation allows the resolution of nmf problems in a recursive fashion we also prove that the underapproximation problem is np hard for any fixed factorization rank using a reduction of the maximum edge biclique problem in bipartite graphs we test two variants of our underapproximation approach on several standard image datasets and show that they provide sparse part based representations with low reconstruction error our results are comparable and sometimes superior to those obtained by two standard sparse nonnegative matrix factorization techniques

adaptive cordic using parallel angle recoding to accelerate rotations the cordic algorithm is used in the evaluation of a wide variety of elementary functions it is a simple and elegant method but it suffers from long latency the angle recoding method is able to reduce the number of iterations by more than 50 percent but its implementation in hardware requires a large increase in cycle time to accommodate its complex angle selection function this restricts its use to those cases where the angle of rotation is fixed and known in advance so that the angle selection can be performed offline this paper presents a simpler implementation of the angle selection scheme that does not require an increase in cycle time thus allowing the angle recoding method to be used dynamically for arbitrary angles the method also has the advantage that all the angle constants are found in parallel in a single step by testing only the initial rotation angle without having to perform successive cordic iterations this dynamic angle recoding method can be formulated to use sections to limit the number of range comparators needed to a reasonable value there is an increase in the number of adaptive cordic iterations needed but this problem can be mitigated by using a buffer in conjunction with the method of sections

smallest singular value of random matrices and geometry of random polytopes abstract we study the behaviour of the smallest singular value of a rectangular random matrix i e matrix whose entries are independent random variables satisfying some additional conditions we prove a deviation inequality and show that such a matrix is a good isomorphism on its image then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes absolutely convex hulls of rows of random matrices all our results hold with high probability that is with probability exponentially in dimension close to 1

implementation of four common functions on an lns co processor we propose a scheme for evaluating four commonly used functions namely 1 inverse trigonometric functions 2 trigonometric functions 3 the exponential function and 4 the logarithmic function with the help of a logarithmic number system lns processor a novel idea of series folding has been introduced for computing the above functions expressed in the form of infinite series we also show that with a suitable choice of the radix for the lns we can evaluate exponential and logarithmic functions without using any extra hardware

power and area efficient unified computation of vector and elementary functions for handheld 3d graphics systems a unified computation method of vector and elementary functions is proposed for handheld 3d graphics systems it unifies vector operations like vector multiply multiply and add divide divide by square root and dot product and elementary functions like trigonometric inverse trigonometric hyperbolic inverse hyperbolic power xy with two variables and logarithm to an arbitrary base into a single four way arithmetic platform a number system called the fixed point hybrid number system fxp hns which combines the fixed point number system fxp and the logarithmic number system lns is proposed for the power and area efficient unification power and area efficient logarithmic and antilogarithmic conversion schemes are also proposed for the data conversions between fixed point and logarithmic numbers in the fxp hns and achieve 0 41 percent and 0 08 percent maximum conversion errors respectively the unified arithmetic unit based on the proposed schemes is presented with less than 6 3 percent operation error its fully pipelined architecture achieves single cycle throughput with maximum four cycle latency for all of the supported operations comparison results show that the proposed arithmetic unit achieves 30 percent power and 10 9 percent area reductions and runs two times faster than the previous approach

the circular law for random matrices we consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance we prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries we assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices the results are based on previous work of bai rudelson and the authors extending results to a larger class of sparse matrices

high speed double precision computation of reciprocal division square root and inverse square root a new method for the high speed computation of double precision floating point reciprocal division square root and inverse square root operations is presented in this paper this method employs a second degree minimax polynomial approximation to obtain an accurate initial estimate of the reciprocal and the inverse square root values and then performs a modified goldschmidt iteration the high accuracy of the initial approximation allows us to obtain double precision results by computing a single goldschmidt iteration significantly reducing the latency of the algorithm two unfolded architectures are proposed the first one computing only reciprocal and division operations and the second one also including the computation of square root and inverse square root the execution times and area costs for both architectures are estimated and a comparison with other multiplicative based methods is presented the results of this comparison show the achievement of a lower latency than these methods with similar hardware requirements

fast evaluation of the elementary functions in single precision in this paper we introduce a new method for the fast evaluation of the elementary functions in single precision based on the evaluation of truncated taylor series using a difference method we assume the availability of large and fast at least for read purposes memory we call this method the ata add table lookup add method as the name implies the hardware required for the method are adders both two and multi operand adders and fast tables for ieee single precision numbers our initial estimates indicate that we can calculate the basic elementary functions namely reciprocal square root logarithm exponential trigonometric and inverse trigonometric functions within the latency of two to four floating point multiplies

algorithm and architecture for logarithm exponential and powering computation an architecture for the computation of logarithm exponential and powering operations is presented in this paper based on a high radix composite algorithm for the computation of the powering function x sup y the algorithm consists of a sequence of overlapped operations 1 digit recurrence logarithm 2 left to right carry free lrcf multiplication and 3 online exponential a redundant number system is used and the selection in 1 and 3 is done by rounding except from the first iteration when selection by table look up is necessary to guarantee the convergence of the recurrences a sequential implementation of the algorithm with a control unit which allows the independent computation of logarithm and exponential is proposed and the execution times and hardware requirements are estimated for single and double precision floating point computations these estimates are obtained for radices from r 8 to r 1 024 according to an approximate model for the delay and area of the main logic blocks and help determining the radix values which lead to the most efficient implementations r 32 and r 128

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

unified mitchell based approximation for efficient logarithmic conversion circuit this paper presents a novel method named the unified mitchell based approximation uma to obtain an optimized mitchell based logarithmic conversion circuit for any desired conversion accuracy up to 14 bits uma is the first method that is able to obtain a conversion circuit when a specific accuracy is required in this work we studied and analyzed five design parameters and their impact on accuracy and hardware merits we formulate the hardware model of the error correction circuit in the conversion circuit for performance evaluation given an accuracy requirement the proposed method explores the design space of the five design parameters as the design space is theoretically huge we propose constraints for the range of the parameter values and develop a systematical search algorithm for exploring the design space uma is able to obtain an area delay product optimized circuit for each of the conversion accuracies achieved by the existing mitchell based designs synthesis results in 90 nm cmos technology show that the circuits obtained are comparable or better than the existing mitchell based designs with the same accuracy objective nine of the fifteen circuits obtained achieve better area delay product by more than 50 percent in addition uma is able to obtain circuits for any accuracy from 4 to 14 bits while the best accuracy achieved by the existing mitchell based methods is less than 12 bits

finding the largest low rank clusters with ky fan 2 k norm and ell_1 norm we propose a convex optimization formulation with the ky fan 2 k norm and ell 1 norm to find k largest approximately rank one submatrix blocks of a given nonnegative matrix that has low rank block diagonal structure with noise we analyze low rank and sparsity structures of the optimal solutions using properties of these two matrix norms we show that under certain hypotheses with high probability the approach can recover rank one submatrix blocks even when they are corrupted with random noise and inserted into a much larger matrix with other random noise blocks

a hybrid number system processor with geometric and complex arithmetic capabilities the architecture design and performance of a hybrid number system processor are described the processor performs multiplication division square root and square in the logarithmic number system lns domain however the input output addition and subtraction are all executed in the 32 b ieee standard floating point number system with the lns multiplier and pipelined architecture the processor is able to perform the geometric and complex arithmetic very effectively the processor is also shown to compare well to an existing 32 b floating point dsp digital signal processor chip for the same level of cmos technology the performance ratios between the hybrid number system and the floating point processor are shown to be 6 4 1 and 8 1 for division and square root respectively for the complex fft fast fourier transform algorithm the ratio is around 2 1

invertibility of sparse non hermitian matrices abstract we consider a class of sparse random matrices of the form a n i j i j i j 1 n where i j are i i d centered random variables and i j are i i d bernoulli random variables taking value 1 with probability p n and prove a quantitative estimate on the smallest singular value for p n log n n under a suitable assumption on the spectral norm of the matrices this establishes the invertibility of a large class of sparse matrices for p n n with some 0 1 we deduce that the condition number of a n is of order n with probability tending to one under the optimal moment assumption on i j this in particular extends a conjecture of von neumann about the condition number to sparse random matrices with heavy tailed entries in the case that the random variables i j are i i d sub gaussian we further show that a sparse random matrix is singular with probability at most exp c n p n whenever p n is above the critical threshold p n log n n the results also extend to the case when i j have a non zero mean we further find quantitative estimates on the smallest singular value of the adjacency matrix of a directed erd s reyni graph whenever its edge connectivity probability is above the critical threshold log n n

random matrices tail bounds for gaps between eigenvalues gaps or spacings between consecutive eigenvalues are a central topic in random matrix theory the goal of this paper is to study the tail distribution of these gaps in various random matrix models we give the first repulsion bound for random matrices with discrete entries and the first super polynomial bound on the probability that a random graph has simple spectrum along with several applications

on the complexity of nonnegative matrix factorization nonnegative matrix factorization nmf has become a prominent technique for the analysis of image databases text databases and other information retrieval and clustering applications the problem is most naturally posed as continuous optimization in this report we define an exact version of nmf then we establish several results about exact nmf i that it is equivalent to a problem in polyhedral combinatorics ii that it is np hard and iii that a polynomial time local search heuristic exists

inverse littlewood offord theorems and the condition number of random discrete matrices consider a random sum r v r nvn where 771 rin are independently and identically distributed i i d random signs and vi vn are integers the littlewood offord problem asks to maximize concentration probabilities such as p r i ih in this paper we develop an inverse littlewood offord theory somewhat in the spirit of freiman s inverse theory in additive combinatorics which starts with the hypothesis that a concentration probability is large and concludes that almost all of the v vn are efficiently contained in a generalized arithmetic progression as an application we give a new bound on the magnitude of the

minimum mean running time function generation using read only memory high speed hardware function generation using table look up in rom and high speed multiplication is considered the reduced interval of interest a b is split into several large partitions within each large partition the functionf x is evaluated by piecewise polynomials of the same low degree whose coefficients are stored in rom four basic architectures for such a scheme are considered a nonlinear programming problem is solved for determining the optimal partitioning of the interval a b the objective function is the average number of multiplications which takes into account the probability distribution r x 1 x ln for the mantissas of normalized floating point numbers where is the radix of the number system the constraint is the available number of rom words the particular case of f x 1 x and 2 is considered in detail and results are presented including an estimate of the number of rom units required

improved decimal floating point logarithmic converter based on selection by rounding this paper presents the algorithm and architecture of the decimal floating point dfp logarithmic converter based on the digit recurrence algorithm with selection by rounding the proposed approach can compute faithful dfp logarithm results for any one of the three dfp formats specified in the ieee 754 2008 standard in order to optimize the latency for the proposed design we mainly integrate the following novel features 1 using the redundant carry save representation of the data path 2 reducing the number of iterations by determining the number of initial iteration and 3 retiming and balancing the delay of the proposed architecture the proposed architecture is synthesized with stm 90 nm standard cell library and the results show that the critical path delay and the number of clock cycles of the proposed decimal64 logarithmic converter are 1 55 ns 34 4 fo4 and 19 respectively and the total hardware complexity is 43 572 nand2 gates the delay estimation results of the proposed architecture show that its latency is close to that of the binary radix 16 logarithmic converter and that it has a significant decrease on latency compared with a recently published high performance cordic implementation

split invariance principles for stationary processes the results of komlos major and tusnady give optimal wiener approximation of partial sums of i i d random variables and provide an extremely powerful tool in probability and statistical inference recently wu ann probab 35 2007 2294 2320 obtained wiener approximation of a class of dependent stationary processes with finite pth moments 2 0 and liu and lin stochastic process appl 119 2009 249 280 removed the logarithmic factor reaching the komlos major tusnady bound o n 1 p no similar results exist for p 4 and in fact no existing method for dependent approximation yields an a s rate better than o n 1 4 in this paper we show that allowing a second wiener component in the approximation we can get rates near to o n 1 p for arbitrary p 2 this extends the scope of applications of the results essentially as we illustrate it by proving new limit theorems for increments of stochastic processes and statistical tests for short term epidemic changes in stationary processes our method works under a general weak dependence condition covering wide classes of linear and nonlinear time series models and classical dynamical systems

a low latency pipelined 2d and 3d cordic processors the unfolded and pipelined cordic is a high performance hardware element that produces a wide variety of one and two argument functions with high throughput the reduction in delay power and area cost are of significant interest regarding this module due to its high demand for resources the linear approximation to rotation has been proposed to achieve such reductions however the schemes for rotation multiplication and vectoring division complicate the implementation in a single unit in this work we improve the linear approximation scheme leading to a unified implementation for rotation and vectoring where fully parallel tree multipliers are used instead of the second half of cordic iterations we also combine the linear approximation to rotation with the scale factor compensation so that the compensation is concurrently performed with the rotation process we then extend the method to 3d cordic such an extension is not straightforward due to the lack of existing analytical expressions for the convergence of the algorithm a comparison using a rough area time model and synthesis results shows that our proposals may achieve significant reductions in delay with no increase in area in actual implementations

on 0 1 polytopes with many facets there exist n dimensional 0 1 polytopes with as many as cn logn n 4 facets this is our main result it answers a question of komei fukuda and gunter m ziegler c 2001 academic press

very high radix circular cordic vectoring and unified rotation vectoring a very high radix algorithm and implementation for circular cordic is presented we first present in depth the algorithm for the vectoring mode in which the selection of the digits is performed by rounding of the control variable to assure convergence with this kind of selection the operands are prescaled however in the cordic algorithm the coordinate x varies during the execution so several scalings might be needed we show that two scalings are sufficient moreover the compensation of the variable scale factor including the cordic scale factor and the prescaling factors is done by computing the logarithm of the scale factor and performing the compensation by an exponential then we combine in a unified unit the proposed vectoring algorithm and the very high radix rotation algorithm which was previously proposed by the authors we compare with low radix implementations in terms of latency and hardware complexity estimations of the delay for 32 bit precision show a speedup of about two with respect to the radix 4 case with redundant addition this speedup is obtained at the cost of an increase in the hardware complexity which is moderate for the pipelined implementation we also compare at the algorithmic level with other very high radix proposals demonstrating the advantages of our algorithms

hardware implementation trade offs of polynomial approximations and interpolations this paper examines the hardware implementation trade offs when evaluating functions via piecewise polynomial approximations and interpolations for precisions of up to 24 bits in polynomial approximations polynomials are evaluated using stored coefficients polynomial interpolations however require the coefficients to be computed on the fly by using stored function values although it is known that interpolations require less memory than approximations but at the expense of additional computations the trade offs in memory area delay and power consumption between the two approaches have not been examined in detail this work quantitatively analyzes these trade offs for optimized approximations and interpolations across different functions and target precisions hardware architectures for degree 1 and degree 2 approximations and interpolations are described the results show that the extent of memory savings realized by using interpolation is significantly lower than what is commonly believed furthermore experimental results on a field programmable gate array fpga show that for high output precision degree 1 interpolations offer considerable area and power savings over degree 1 approximations but similar savings are not realized when degree 2 interpolations and approximations are compared the availability of both interpolation based and approximation based designs offers a richer set of design trade offs than what is available using either interpolation or approximation alone

high speed function approximation using a minimax quadratic interpolator a table based method for high speed function approximation in single precision floating point format is presented in this paper our focus is the approximation of reciprocal square root square root reciprocal exponentials logarithms trigonometric functions powering with a fixed exponent p or special functions the algorithm presented here combines table look up an enhanced minimax quadratic approximation and an efficient evaluation of the second degree polynomial using a specialized squaring unit redundant arithmetic and multioperand addition the execution times and area costs of an architecture implementing our method are estimated showing the achievement of the fast execution times of linear approximation methods and the reduced area requirements of other second degree interpolation algorithms moreover the use of an enhanced minimax approximation which through an iterative process takes into account the effect of rounding the polynomial coefficients to a finite size allows for a further reduction in the size of the look up tables to be used making our method very suitable for the implementation of an elementary function generator in state of the art dsps or graphics processing units gpus

hilbert space representations of m dependent processes a representation of one dependent processes is given in terms of hilbert spaces vectors and bounded linear operators on hilbert spaces this generalizes a construction of one dependent processes that are not two block factors we show that all one dependent processes admit a representation we prove that if there is in the hilbert space a closed convex cone that is invariant under certain operators and that is spanned by a finite number of linearly independent vectors then the corresponding process is a two block factor of an independent process apparently the difference between two block factors and non two block factors is determined by the geometry of invariant cones the dimension of the smallest hilbert space that represents a process is a measure for the complexity of the structure of the process for two valued one dependent processes if there is a cylinder with measure equal to zero then this process can be represented by a hilbert space with dimension smaller than or equal to the length of this cylinder in the two valued case a cylinder with measure equal to zero whose length is minimal and less than or equal to 7 is symmetric we generalize the concept of hilbert space representation to m dependent processes and it turns out that all m dependent processes admit a representation several theorems can be generalized to m dependent processes

asymptotics for high dimensional regression m estimates fixed design results we investigate the asymptotic distributions of coordinates of regression m estimates in the moderate p n regime where the number of covariates p grows proportionally with the sample size n under appropriate regularity conditions we establish the coordinate wise asymptotic normality of regression m estimates assuming a fixed design matrix our proof is based on the second order poincare inequality and leave one out analysis some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices we also show a counterexample namely an anova type design to emphasize that the technical assumptions are not just artifacts of the proof finally numerical experiments confirm and complement our theoretical results

poincare inequality in mean value for gaussian polytopes let k n g 1 g n be the absolute convex hull of n independent standard gaussian random points in mathbb r n with n n we prove that for any 1 lipschitz function f mathbb r n rightarrow mathbb r the polytope k n satisfies the following poincare inequality in mean value mathbb e omega int limits k n omega left f x frac 1 textup vol n left k n omega right int limits k n omega f y dy right 2 dx leq frac c n mathbb e omega int limits k n omega x 2dx where c 0 is an absolute constant this poincare inequality is the one suggested by a conjecture of kannan lovasz and simonovits for general convex bodies moreover we prove in mean value that the volume of the polytope k n is concentrated in a subexponential way within a thin euclidean shell with the optimal dependence of the dimension n an important tool of the proofs is a representation of the law of g 1 g n conditioned by the event that the convex hull of g 1 g n is a n 1 face of k n as an application we also get an estimate of the number of n 1 faces of the polytope k n valid for every n n

euclidean embeddings in spaces of finite volume ratio via random matrices let rn be the space rn equipped with a norm whose unit ball has a bounded volume ratio with respect to the euclidean unit ball let be any random n n matrix with n n whose entries are independent random variables satisfying some moment assumptions we show that with high probability is a good isomorphism from the n dimensional euclidean space rn onto its image in rn i e there exist 0 such that for all x rn n x x n x this solves a conjecture of schechtman on random embeddings of 2 into n 1

a unified architecture for the accurate and high throughput implementation of six key elementary functions this paper presents a unified architecture for the compact implementation of several key elementary functions including reciprocal square root and logarithm in single precision floating point arithmetic the proposed high throughput design is based on uniform domain segmentation and curve fitting techniques numerically accurate least squares regression is utilized to calculate the polynomial coefficients the architecture is optimized by analyzing the trade off between the size of the required memory and the precision of intermediate variables to achieve the minimum 23 bit accuracy required for single precision floating point representation the efficiency of the proposed unified data path is demonstrated on a common field programmable gate array

on the geometry of random convex sets between polytopes and zonotopes abstract in this work we study a class of random convex sets that interpolate between polytopes and zonotopes these sets arise from considering a q t h moment q 1 of an average of order statistics of 1 dimensional marginals of a sequence of n n independent random vectors in r n we consider the random model of isotropic log concave distributions as well as the uniform distribution on an l p n sphere 1 p with respect to the cone probability measure and study the geometry of these sets in terms of the support function and mean width we provide asymptotic formulas for the expectation of these geometric functionals which are sharp up to absolute constants our model includes and generalizes the standard one for random polytopes

hardware designs for exactly rounded elementary functions this paper presents hardware designs that produce exactly rounded results for the functions of reciprocal square root 2 sup x and log sub 2 x these designs use polynomial approximation in which the terms in the approximation are generated in parallel and then summed by using a multi operand adder to reduce the number of terms in the approximation the input interval is partitioned into subintervals of equal size and different coefficients are used for each subinterval the coefficients used in the approximation are initially determined based on the chebyshev series approximation they are then adjusted to obtain exactly rounded results for all inputs hardware designs are presented and delay and area comparisons are made based on the degree of the approximating polynomial and the accuracy of the final result for single precision floating point numbers a design that produces exactly rounded results for all four functions has an estimated delay of 80 ns and a total chip area of 98 mm sup 2 in a 1 0 micron cmos technology allowing the results to have a maximum error of one unit in the last place reduces the computational delay by 5 to 30 and the area requirements by 33 to 77

toward correctly rounded transcendentals the table maker s dilemma is the problem of always getting correctly rounded results when computing the elementary functions after a brief presentation of this problem we present new developments that have helped us to solve this problem for the double precision exponential function in a small domain these new results show that this problem can be solved at least for the double precision format for the most usual functions

finding approximately rank one submatrices with the nuclear norm and ell_1 norm we propose a convex optimization formulation with the nuclear norm and ell 1 norm to find a large approximately rank one submatrix of a given nonnegative matrix we develop optimality conditions for the formulation and characterize the properties of the optimal solutions we establish conditions under which the optimal solution of the convex formulation has a specific sparse structure finally we show that under certain hypotheses with high probability the approach can recover the rank one submatrix even when it is corrupted with random noise and inserted as a submatrix into a much larger random noise matrix

extremal two correlations of two valued stationary one dependent processes the maximal value of the two correlation for two valued stationary one dependent processes with fixed probability of a single symbol is determined we show that the process attaining this bound is unique except when 1 2 when there are exactly two different processes the analogous problem for minimal two correlation is discussed and partial results are obtained

radix 4 reciprocal square root and its combination with division and square root in this work we present a reciprocal square root algorithm by digit recurrence and selection by a staircase function and the radix 4 implementation as in similar algorithms for division and square root the results are obtained correctly rounded in a straightforward manner in contrast to existing methods to compute the reciprocal square root although apparently a single selection function can only be used for j spl ges 2 the selection constants are different for j 0 j 1 and j spl ges 2 we show that it is possible to use a single selection function for all iterations we perform a rough comparison with existing methods and we conclude that our implementation is a low hardware complexity solution with moderate latency especially for exactly rounded results we also extend the unit to support division and square root with the same selection function and with slight modifications in the initialization of the reciprocal square root unit

asymptotic expansions for potential functions of i i d random fields for sums of finite range potential functions of an iid random field we derive the validity of formal expansions of length two under standard conditions formal expansions are valid if and only if the characteristic functions of the sum converge to zero for all nonzero frequency parameters if this convergence fails the distribution of the sum can be approximated by a mixture of lattice distributions the result applies to m dependent random fields generated by independent random variables

low power logarithmic number system addition subtraction and their impact on digital filters this paper presents techniques for low power addition subtraction in the logarithmic number system lns and quantifies their impact on digital filter vlsi implementation the impact of partitioning the look up tables required for lns addition subtraction on complexity performance and power dissipation of the corresponding circuits is quantified two design parameters are exploited to minimize complexity namely the lns base and the organization of the lns word a roundoff noise model is used to demonstrate the impact of base and word length on the signal to noise ratio of the output of finite impulse response fir filters in addition techniques for the low power implementation of an lns multiply accumulate mac units are investigated furthermore it is shown that the proposed techniques can be extended to cotransformation based circuits that employ interpolators the results are demonstrated by evaluating the power dissipation complexity and performance of several fir filter configurations comprising one two or four mac units simulations of placed and routed vlsi lns based digital filters using a 90 nm 1 0 v cmos standard cell library reveal that significant power dissipation savings are possible by using optimized lns circuits at no performance penalty when compared to linear fixed point two s complement equivalents

fast hardware based algorithms for elementary function computations using rectangular multipliers as the name suggests elementary functions play a vital role in scientific computations yet due to their inherent nature they are a considerable computing task by themselves not surprisingly since the dawn of computing the goal of speeding up elementary function computation has been pursued this paper describes new hardware based algorithms for the computation of the common elementary functions namely division logarithm reciprocal square root arc tangent sine and cosine these algorithms exploit microscopic parallelism using specialized hardware with heavy use of truncation based on detailed accuracy analysis the contribution of this work lies in the fact that these algorithms are very fast and yet are accurate if we let the time to perform an ieee standard 754 double precision floating point multiplication be spl tau sub spl times our algorithms to achieve roughly 3 68 spl tau sub spl times 4 56 spl tau sub spl times 5 25 spl tau sub spl times 3 69 spl tau sub spl times 7 06 spl tau sub spl times and 6 5 spl tau sub spl times for division logarithm square root exponential are tangent and complex exponential sine and cosine respectively the trade off is the need for tables and some specialized hardware the total amount of tables required however is less than 128 kbytes we discuss the hardware algorithmic and accuracy aspects of these algorithms

elementary functions hardware implementation using constrained piecewise polynomial approximations a novel technique for designing piecewise polynomial interpolators for hardware implementation of elementary functions is investigated in this paper in the proposed approach the interval where the function is approximated is subdivided in equal length segments and two adjacent segments are grouped in a segment pair suitable constraints are then imposed between the coefficients of the two interpolating polynomials in each segment pair this allows reducing the total number of stored coefficients it is found that the increase in the approximation error due to constraints between polynomial coefficients can easily be overcome by increasing the fractional bits of the coefficients overall compared with standard unconstrained piecewise polynomial approximation having the same accuracy the proposed method results in a considerable advantage in terms of the size of the lookup table needed to store polynomial coefficients the calculus of the coefficients of constrained polynomials and the optimization of coefficients bit width is also investigated in this paper results for several elementary functions and target precision ranging from 12 to 42 bits are presented the paper also presents vlsi implementation results targeting a 90 nm cmos technology and using both direct and horner architectures for constrained degree 1 degree 2 and degree 3 approximations

cmos vlsi implementation of a low power logarithmic converter we present a unique 32 bit binary to binary logarithm converter including its cmos vlsi implementation the converter is implemented using combinational logic only and it calculates a logarithm approximation in a single clock cycle unlike other complex logarithm correcting algorithms three unique algorithms are developed and implemented with low power and fast circuits that reduce the maximum percent errors that result from binary to binary logarithm conversion to 0 9299 percent 0 4314 percent and 0 1538 percent fast 4 16 and 32 bit leading one detector circuits are designed to obtain the leading one position of an input binary word a 32 word spl times 5 bit mos rom is used to provide 5 bit integers based on the corresponding leading one position both converter area and speed have been considered in the design approach resulting in the use of a very efficient 32 bit logarithmic shifter in the 32 bit logarithmic converter the converter is implemented using 0 6 spl mu m cmos technology and it requires 1 600 spl lambda spl times 2 800 spl lambda of chip area simulations of the cmos design for the 32 bit logarithmic converter operating at v sub dd equal to 5 volts run at 55 mhz and the converter consumes 20 milliwatts

efficient techniques for binary to multidigit multidimensional logarithmic number system conversion using range addressable look up tables the multidimensional logarithmic number system mdlns which has similar properties to the classical logarithmic number system lns provides more degrees of freedom than the lns by virtue of having two or more orthogonal bases and has the ability to use multiple mdlns components or digits unlike the lns there is no monotonic relationship between standard binary representations and mdlns representations using look up tables luts to perform the mapping function can be unrealistic for hardware implementations when large binary ranges or multiple digits are used this work proposes a novel range addressable technique for using look up tables that allows efficient conversion from binary to single or multidigit mdlns with varying accuracies depending on the selected implementation

near optimal sample complexity bounds for robust learning of gaussian mixtures via compression schemes we introduce a novel technique for distribution learning based on a notion of sample compression any class of distributions that allows such a compression scheme can be learned with few samples m

improving accuracy in mitchell s logarithmic multiplication using operand decomposition logarithmic number systems lns offer a viable alternative in terms of area delay and power to binary number systems for implementing multiplication and division operations for applications in signal processing the mitchell algorithm ma proposed reduces the complexity of finding the logarithms and the antilogarithms using piecewise straight line approximations of the logarithm and the antilogarithm curves the approximations however result in some loss of accuracy thus several methods have been proposed in the literature for improving the accuracy of mitchell s algorithm in this work we investigate a new method based on operand decomposition od to improve the accuracy of mitchell s algorithm when applied to logarithmic multiplication in the od technique proposed for reducing the amount of switching activity in binary multiplication the two inputs to be multiplied are together decomposed into four binary operands and the product is expressed as the sum of the products of the decomposed numbers we show that applying operand decomposition to the inputs as a preprocessing step to mitchell s multiplication algorithm significantly improves the accuracy experimental results indicate that the proposed algorithm for logarithmic multiplication reduces the error percentage of mitchell s algorithm by 44 7 percent on the average it is also shown that the od method yields further improvement when combined with the other correction methods proposed in the literature

properties of linear approximations of matrices in the spectral norm abstract the properties of linear approximations of a matrix are presented with respect to the spectrum norm we describe the connection between all spectral approximations of a given matrix and we give a sufficient condition under which the spectral approximation is unique

a hardwired generalized algorithm for generating the logarithm base k by iteration a number of previous works of computing the binary logarithm have been developed by a series of line approximations with moderate errors 1 3 in this paper a generalized iterative method for the logarithm of base k evalution is proposed we suggest that an integral constant k is used as a comparative index in each iteration no table lookup is required it can yield a more accurate and rapidly convergent value and can be easily implemented either by software or a hardwired device a numerical example is illustrated for the process of calculation the accuracy is analyzed and plotted by using two schemes in which eight iteration cycle and 16 iteration cycle length are used respectively

an algebraic construction of a class of one dependent processes a special class of stationary one dependent two valued stochastic processes is defined we associate to each member of this class two parameter values whereby different members receive different parameter values for any given values of the parameters we show how to determine whether 1 a process exists having the given parameter values and if so 2 this process can be obtained as a two block factor from an independent process this determines a two parameter subfamily of the class of stationary one dependent two valued stochastic processes which are not two block factors of independent processes

sign logarithm arithmetic for fft implementation sign logarithm arithmetic is applicable to a variety of numerical applications where wide dynamic range and small wordsize are required in this paper the basic sign logarithm arithmetic operations required for signal processing i e addition subtraction and multiplication are reviewed the computational errors are analyzed for fft realization and simulation results are presented which serve to verify the analysis it is shown that the sign logarithm approach provides improved arithmetic quantization error performance for a given word size over fft s implemented with conventional fixed or floating point arithmetic and that the sign logarithm implementation is faster and less complex than conventional approaches

lower bounds on nonnegative rank via nonnegative nuclear norms the nonnegative rank of an entrywise nonnegative matrix a in mathbb r m times n a r m n is the smallest integer r r such that a a can be written as a uv a uv where u in mathbb r m times r u r m r and v in mathbb r r times n v r r n are both nonnegative the nonnegative rank arises in different areas such as combinatorial optimization and communication complexity computing this quantity is np hard in general and it is thus important to find efficient bounding techniques especially in the context of the aforementioned applications in this paper we propose a new lower bound on the nonnegative rank which unlike most existing lower bounds does not solely rely on the matrix sparsity pattern and applies to nonnegative matrices with arbitrary support the idea involves computing a certain nuclear norm with nonnegativity constraints which allows to lower bound the nonnegative rank in the same way the standard nuclear norm gives lower bounds on the standard rank our lower bound is expressed as the solution of a copositive programming problem and can be relaxed to obtain polynomial time computable lower bounds using semidefinite programming we compare our lower bound with existing ones and we show examples of matrices where our lower bound performs better than currently known ones

a low error cost efficient design procedure for evaluating logarithms to be used in a logarithmic arithmetic processor based on an error flattened non uniform region linear approximation algorithm this brief proposes a low error and a cost efficient design procedure for realizing an optimized shift and add logarithmic unit lu which uses minimum hardware to meet the desired error constraint for embedded graphics systems mathematically this brief first derives two solutions of the error flattened algorithm subsequently for an error constraint the minimum number of approximation regions n the corresponding i th interpolation coefficients ai bi and the regional endpoints xi 1 xi 1 i n are obtained accordingly using the unique properties of the logarithmic function spacing of xi is non uniform to make errors in each region consistent carefully examining the cost of the applied add sub network a low cost candidate for a shift and add lu is determined next a cost exploration process which gradually increases n is performed a large number of regions results in a more accurate conversion algorithm that might tolerate more implementation errors by using simple hardware whose cost is lower than that of the inferior candidate after exploring the cost based on error tolerance the proposed design procedure finally generates a hardware to meet the desired error constraint slightly modifying the regional endpoint xi increases hardware efficiency at the cost of increased error proposed circuits were synthesized in umc 65rvt cmos technology compared to state of the art shift and add logarithmic converters simulation results reveal that the proposed design saves approximately 12 7 51 1 percent area of polynomial approximation and improves approximately 1 14 db snr gain while achieving a tighter error constraint

interleaved memory function interpolators with application to an accurate lns arithmetic unit this paper describes a new method for polynomial interpolation in hardware with advantages demonstrated by its application to an accurate logarithmic number system lns arithmetic unit the use of an interleaved memory reduces storage requirements by allowing each stored function value to be used in interpolation across several segments this strategy can be shown to always use fewer words of memory than an optimized polynomial with stored polynomial coefficients interleaved memory function interpolators are then applied to the specific goal of an accurate logarithmic number system arithmetic unit many accuracy requirements for the lns arithmetic unit are possible although a round to nearest would be desirable it cannot be easily achieved the goal suggested is to insure that the worst case lns relative error is smaller than the worst case floating point fp relative error using the interleaved memory interpolator the detailed design of an lns arithmetic unit is performed using a second order polynomial interpolator including approximately 91k bits of rom this arithmetic unit has better accuracy and less complexity than previous lns units

self scaled bounds for atomic cone ranks applications to nonnegative rank and cp rank the nonnegative rank of a matrix a is the smallest integer r such that a can be written as the sum of r rank one nonnegative matrices the nonnegative rank has received a lot of attention recently due to its application in optimization probability and communication complexity in this paper we study a class of atomic rank functions defined on a convex cone which generalize several notions of positive ranks such as nonnegative rank or cp rank for completely positive matrices the main contribution of the paper is a new method to obtain lower bounds for such ranks additionally the bounds we propose can be computed by semidefinite programming using sum of squares relaxations the idea of the lower bound relies on an atomic norm approach where the atoms are self scaled according to the vector or matrix in the case of nonnegative rank of interest this results in a lower bound that is invariant under scaling and that enjoys other interesting structural properties for the case of the nonnegative rank we show that our bound has an appealing connection with existing combinatorial bounds and other norm based bounds for example we show that our lower bound is a non combinatorial version of the fractional rectangle cover number while the sum of squares relaxation is closely related to the lovasz bar vartheta number of the rectangle graph of the matrix we also prove that the lower bound is always greater than or equal to the hyperplane separation bound and other similar norm based bounds we also discuss the case of the tensor nonnegative rank as well as the cp rank and compare our bound with existing results

diameters of sections and coverings of convex bodies abstract we study the diameters of sections of convex bodies in r n determined by a random n n matrix either as kernels of or as images of entries of are independent random variables satisfying some boundedness conditions and typical examples are matrices with gaussian or bernoulli random variables we show that if a symmetric convex body k in r n has one well bounded k codimensional section then for any m ck random sections of k of codimension m are also well bounded where c 1 is an absolute constant it is noteworthy that in the gaussian case when determines randomness in sense of the haar measure on the grassmann manifold we can take c 1

spectral norm of products of random and deterministic matrices we study the spectral norm of matrices w that can be factored as w ba where a is a random matrix with independent mean zero entries and b is a fixed matrix under the 4 e th moment assumption on the entries of a we show that the spectral norm of such an m n matrix w is bounded by sqrt m sqrt n which is sharp in other words in regard to the spectral norm products of random and deterministic matrices behave similarly to random matrices with independent entries this result along with the previous work of rudelson and the author implies that the smallest singular value of a random m n matrix with i i d mean zero entries and bounded 4 e th moment is bounded below by sqrt m sqrt n 1 with high probability

poisson polyhedra in high dimensions abstract the zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated as the space dimension tends to infinity the model includes the zero cell of stationary and isotropic poisson hyperplane tessellations as well as the typical cell of a stationary poisson voronoi tessellation as special cases it is shown that asymptotically in the space dimension with overwhelming probability these cells satisfy the hyperplane conjecture if the distance exponent and the intensity parameter are suitably chosen dimension dependent functions also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed in the background are new identities linking the f vector of the zero cell to certain dual intrinsic volumes

approximating elementary functions with symmetric bipartite tables this paper presents a high speed method for function approximation that employs symmetric bipartite tables this method performs two parallel table lookups to obtain a carry save borrow save function approximation which is either converted to a two s complement number or is booth encoded compared to previous methods for bipartite table approximations this method uses less memory by taking advantage of symmetry and leading zeros in one of the two tables it also has a closed form solution for the table entries provides tight bounds on the maximum absolute error and can be applied to a wide range of functions a variation of this method provides accurate initial approximations that are useful in multiplicative divide and square root algorithms

reciprocation square root inverse square root and some elementary functions using small multipliers this paper deals with the computation of reciprocals square roots inverse square roots and some elementary functions using small tables small multipliers and for some functions a final large almost full length multiplication we propose a method based on argument reduction and series expansion that allows fast evaluation of these functions in high precision the strength of this method is that the same scheme allows the computation of all these functions we estimate the delay the size number of tables and the size number of multipliers and compare with other related methods

discussion latent variable graphical model selection via convex optimization since recently there have been an increasing interest in the problem of estimating a high dimensional matrix k that can be decomposed in a sum of a sparse matrix s i e a matrix having only a small number of nonzero entries and a low rank matrix l this is motivated by applications in computer vision video segmentation computational biology semantic indexing etc the main contribution and novelty of chandrasekaran parrilo and willsky paper cpw in what follows is to propose and study a method of inference about such decomposable matrices for a particular setting where k is the precision concentration matrix of a partially observed sparse gaussian graphical model ggm in this case k is the inverse of the covariance matrix of a gaussian vector xo extracted from a larger gaussian vector xo xh with sparse inverse covariance matrix then it is easy to see that k can be represented as a sum of a sparse precision matrix s corresponding to the observed variables xo and a matrix l with rank at most h where h is the dimension of the latent variables xh if h is small which is a typical situation in practice then l has low rank the ggm with latent variables is of major interest for applications in biology or in social networks where one often does not observe all the variables relevant for depicting sparsely the conditional dependencies note that formally this is just one possible motivation and mathematically the problem is dealt with in more generality namely postulating that the precision matrix satisfies