One popular method for dealing with large-scale data sets is sampling. For example, by using the empirical statistical leverage scores as an importance sampling distribution, the method of algorithmic leveraging samples and rescales rows columns of data matrices to reduce the data size before performing computations on the subproblem. This method has been successful in improving computational efficiency of algorithms for matrix problems such as least-squares approximation, least absolute deviations approximation, and low-rank matrix approximation. Existing work has focused on algorithmic issues such as worst-case running times and numerical issues associated with providing high-quality implementations, but none of it addresses statistical aspects of this method. In this paper, we provide a simple yet effective framework to evaluate the statistical properties of algorithmic leveraging in the context of estimating parameters in a linear regression model with a fixed number of predictors. We show that from the statistical perspective of bias and variance, neither leverage-based sampling nor uniform sampling dominates the other. This result is particularly striking, given the well-known result that, from the algorithmic perspective of worst-case analysis, leverage-based sampling provides uniformly superior worst-case algorithmic results, when compared with uniform sampling. Based on these theoretical results, we propose and analyze two new leveraging algorithms. A detailed empirical evaluation of existing leverage-based methods as well as these two new methods is carried out on both synthetic and real data sets. The empirical results indicate that our theory is a good predictor of practical performance of existing and new leverage-based algorithms and that the new algorithms achieve improved performance.
Although Uniform Subsampling (with or without replacement) is very simple to implement, it is easy to construct examples where it will perform very poorly (e.g., see below or see @cite_19 @cite_32 ). On the other hand, it has been shown that, for a parameter @math to be tuned, if then the following relative-error bounds hold: where @math is the condition number of @math and where @math is a parameter defining the amount of the mass of @math inside the column space of @math @cite_19 @cite_32 @cite_1 .
@cite_19 @cite_1 @cite_32
2007385254 2099641086 2950933581 2145096794
We initiate a study of when the value of mathematical relaxations such as linear and semi-definite programs for constraint satisfaction problems (CSPs) is approximately preserved when restricting the instance to a sub-instance induced by a small random subsample of the variables. Let C be a family of CSPs such as 3SAT, Max-Cut, etc., and let Π be a mathematical program that is a relaxation for C, in the sense that for every instance P ∈ C, Π(P) is a number in [0, 1] upper bounding the maximum fraction of satisfiable constraints of P. Loosely speaking, we say that subsampling holds for C and Π if for every sufficiently dense instance P ∈ C and every e > 0, if we let P' be the instance obtained by restricting P to a sufficiently large constant number of variables, then Π(P') ∈ (1 ± e)Π(P). We say that weak subsampling holds if the above guarantee is replaced with Π(P') = 1 − θ(γ) whenever Π(P) = 1 − γ, where θ hides only absolute constants. We obtain both positive and negative results, showing that: 1. Subsampling holds for the BasicLP and BasicSDP programs. BasicSDP is a variant of the semi-definite program considered by Raghavendra (2008), who showed it gives an optimal approximation factor for every constraint-satisfaction problem under the unique games conjecture. BasicLP is the linear programming analog of BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of unique games type. 3. There are non-unique CSPs for which even weak subsampling fails for the above tighter semi-definite programs. Also there are unique CSPs for which (even weak) subsampling fails for the Sherali-Adams linear programming hierarchy. As a corollary of our weak subsampling for strong semi-definite programs, we obtain a polynomial-time algorithm to certify that random geometric graphs (of the type considered by Feige and Schechtman, 2002) of max-cut value 1 − γ have a cut value at most 1 − γ 10. More generally, our results give an approach to obtaining average-case algorithms for CSPs using semi-definite programming hierarchies. Suppose a discrete-time signal S(t), 0 spl les t<N, is a superposition of atoms taken from a combined time-frequency dictionary made of spike sequences 1 sub t= spl tau and sinusoids exp 2 spl pi iwt N spl radic N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discrete-time signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time-frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the l sup 1 norm of the coefficients among all decompositions. Here "highly sparse" means that N sub t +N sub w < spl radic N 2 where N sub t is the number of time atoms, N sub w is the number of frequency atoms, and N is the length of the discrete-time signal. Underlying this result is a general l sup 1 uncertainty principle which says that if two bases are mutually incoherent, no nonzero signal can have a sparse representation in both bases simultaneously. For the above setting, the bases are sinusoids and spikes, and mutual incoherence is measured in terms of the largest inner product between different basis elements. The uncertainty principle holds for a variety of interesting basis pairs, not just sinusoids and spikes. The results have idealized applications to band-limited approximation with gross errors, to error-correcting encryption, and to separation of uncoordinated sources. Related phenomena hold for functions of a real variable, with basis pairs such as sinusoids and wavelets, and for functions of two variables, with basis pairs such as wavelets and ridgelets. In these settings, if a function f is representable by a sufficiently sparse superposition of terms taken from both bases, then there is only one such sparse representation; it may be obtained by minimum l sup 1 norm atomic decomposition. The condition "sufficiently sparse" becomes a multiscale condition; for example, that the number of wavelets at level j plus the number of sinusoids in the jth dyadic frequency band are together less than a constant times 2 sup j 2 . We investigate the approximability of several classes of real-valued functions by functions of a small number of variables ( juntas ). Our main results are tight bounds on the number of variables required to approximate a function @math within @math -error @math over the uniform distribution: 1. If @math is submodular, then it is @math -close to a function of @math variables. This is an exponential improvement over previously known results. We note that @math variables are necessary even for linear functions. 2. If @math is fractionally subadditive (XOS) it is @math -close to a function of @math variables. This result holds for all functions with low total @math -influence and is a real-valued analogue of Friedgut's theorem for boolean functions. We show that @math variables are necessary even for XOS functions. As applications of these results, we provide learning algorithms over the uniform distribution. For XOS functions, we give a PAC learning algorithm that runs in time @math . For submodular functions we give an algorithm in the more demanding PMAC learning model (Balcan and Harvey, 2011) which requires a multiplicative @math factor approximation with probability at least @math over the target distribution. Our uniform distribution algorithm runs in time @math . This is the first algorithm in the PMAC model that over the uniform distribution can achieve a constant approximation factor arbitrarily close to 1 for all submodular functions. As follows from the lower bounds in (, 2013) both of these algorithms are close to optimal. We also give applications for proper learning, testing and agnostic learning with value queries of these classes. This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f spl isin C sup N and a randomly chosen set of frequencies spl Omega . Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set spl Omega ? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)= spl sigma sub spl tau spl isin T f( spl tau ) spl delta (t- spl tau ) obeying |T| spl les C sub M spl middot (log N) sup -1 spl middot | spl Omega | for some constant C sub M >0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N sup -M ), f can be reconstructed exactly as the solution to the spl lscr sub 1 minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C sub M which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T| spl middot logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N sup -M ) would in general require a number of frequency samples at least proportional to |T| spl middot logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.