We study the estimation error of constrained M-estimators, and derive explicit upper bounds on the expected estimation error determined by the Gaussian width of the constraint set. Both of the cases where the true parameter is on the boundary of the constraint set (matched constraint), and where the true parameter is strictly in the constraint set (mismatched constraint) are considered. For both cases, we derive novel universal estimation error bounds for regression in a generalized linear model with the canonical link function. Our error bound for the mismatched constraint case is minimax optimal in terms of its dependence on the sample size, for Gaussian linear regression by the Lasso.
We say that the constraint is if @math lies on the boundary of @math in ) (or @math ), and if @math lies strictly in @math (or @math ). The analyses in @cite_20 @cite_7 require the constraint to be matched, while in practice the exact value of @math is seldom known. The constraint in @cite_2 is always matched due to the special structure of quantum density operators. The error bounds in @cite_24 @cite_0 can be overly pessimistic, because they hold for all @math . The results in @cite_6 @cite_23 @cite_10 do not require a matched constraint and depend on @math ; our result is of this kind. Recall that, however, @cite_6 is limited to specific statistical models, and @cite_23 @cite_10 are limited to specific @math -estimators.
@cite_7 @cite_6 @cite_24 @cite_0 @cite_23 @cite_2 @cite_10 @cite_20
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We consider the problem of estimating an unknown signal @math from noisy linear observations @math . In many practical instances, @math has a certain structure that can be captured by a structure inducing convex function @math . For example, @math norm can be used to encourage a sparse solution. To estimate @math with the aid of @math , we consider the well-known LASSO method and provide sharp characterization of its performance. We assume the entries of the measurement matrix @math and the noise vector @math have zero-mean normal distributions with variances @math and @math respectively. For the LASSO estimator @math , we attempt to calculate the Normalized Square Error (NSE) defined as @math as a function of the noise level @math , the number of observations @math and the structure of the signal. We show that, the structure of the signal @math and choice of the function @math enter the error formulae through the summary parameters @math and @math , which are defined as the Gaussian squared-distances to the subdifferential cone and to the @math -scaled subdifferential, respectively. The first LASSO estimator assumes a-priori knowledge of @math and is given by @math . We prove that its worst case NSE is achieved when @math and concentrates around @math . Secondly, we consider @math , for some @math . This time the NSE formula depends on the choice of @math and is given by @math . We then establish a mapping between this and the third estimator @math . Finally, for a number of important structured signal classes, we translate our abstract formulae to closed-form upper bounds on the NSE. We obtain sharp bounds on the performance of Empirical Risk Minimization performed in a convex class and with respect to the squared loss, without assuming that class members and the target are bounded functions or have rapidly decaying tails. Rather than resorting to a concentration-based argument, the method used here relies on a small-ball' assumption and thus holds for classes consisting of heavy-tailed functions and for heavy-tailed targets. The resulting estimates scale correctly with the noise level' of the problem, and when applied to the classical, bounded scenario, always improve the known bounds. This paper develops theoretical results regarding noisy 1-bit compressed sensing and sparse binomial regression. We demonstrate that a single convex program gives an accurate estimate of the signal, or coefficient vector, for both of these models. We show that an -sparse signal in can be accurately estimated from m = O(s log(n s)) single-bit measurements using a simple convex program. This remains true even if each measurement bit is flipped with probability nearly 1 2. Worst-case (adversarial) noise can also be accounted for, and uniform results that hold for all sparse inputs are derived as well. In the terminology of sparse logistic regression, we show that O (s log (2n s)) Bernoulli trials are sufficient to estimate a coefficient vector in which is approximately -sparse. Moreover, the same convex program works for virtually all generalized linear models, in which the link function may be unknown. To our knowledge, these are the first results that tie together the theory of sparse logistic regression to 1-bit compressed sensing. Our results apply to general signal structures aside from sparsity; one only needs to know the size of the set where signals reside. The size is given by the mean width of K, a computable quantity whose square serves as a robust extension of the dimension. This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of asymptotic convex geometry, and demonstrates connections between geometric results and estimation problems. The theory is illustrated with applications to sparse recovery, matrix completion, quantization, linear and logistic regression and generalized linear models. We study the problem of signal estimation from non-linear observations when the signal belongs to a low-dimensional set buried in a high-dimensional space. A rough heuristic often used in practice postulates that non-linear observations may be treated as noisy linear observations, and thus the signal may be estimated using the generalized Lasso. This is appealing because of the abundance of efficient, specialized solvers for this program. Just as noise may be diminished by projecting onto the lower dimensional space, the error from modeling non-linear observations with linear observations will be greatly reduced when using the signal structure in the reconstruction. We allow general signal structure, only assuming that the signal belongs to some set K in R^n. We consider the single-index model of non-linearity. Our theory allows the non-linearity to be discontinuous, not one-to-one and even unknown. We assume a random Gaussian model for the measurement matrix, but allow the rows to have an unknown covariance matrix. As special cases of our results, we recover near-optimal theory for noisy linear observations, and also give the first theoretical accuracy guarantee for 1-bit compressed sensing with unknown covariance matrix of the measurement vectors. Author(s): Plan, Y; Vershynin, R; Yudovina, E | Abstract: Consider measuring an n-dimensional vector x through the inner product with several measurement vectors, a_1, a_2, ..., a_m. It is common in both signal processing and statistics to assume the linear response model y_i = + e_i, where e_i is a noise term. However, in practice the precise relationship between the signal x and the observations y_i may not follow the linear model, and in some cases it may not even be known. To address this challenge, in this paper we propose a general model where it is only assumed that each observation y_i may depend on a_i only through . We do not assume that the dependence is known. This is a form of the semiparametric single index model, and it includes the linear model as well as many forms of the generalized linear model as special cases. We further assume that the signal x has some structure, and we formulate this as a general assumption that x belongs to some known (but arbitrary) feasible set K. We carefully detail the benefit of using the signal structure to improve estimation. The theory is based on the mean width of K, a geometric parameter which can be used to understand its effective dimension in estimation problems. We determine a simple, efficient two-step procedure for estimating the signal based on this model -- a linear estimation followed by metric projection onto K. We give general conditions under which the estimator is minimax optimal up to a constant. This leads to the intriguing conclusion that in the high noise regime, an unknown non-linearity in the observations does not significantly reduce one's ability to determine the signal, even when the non-linearity may be non-invertible. Our results may be specialized to understand the effect of non-linearities in compressed sensing. This paper considers the linear inverse problem where we wish to estimate a structured signal x_0 from its corrupted observations. When the problem is ill-posed, it is natural to associate a convex function f(·) with the structure of the signal. For example, l_1 norm can be used for sparse signals. To carry out the estimation, we consider two well-known convex programs: 1) Second order cone program (SOCP), and, 2) Lasso. Assuming Gaussian measurements, we show that, if precise information about the value f(x_0) or the l_2-norm of the noise is available, one can do a particularly good job at estimation. In particular, the reconstruction error becomes proportional to the “sparsity” of the signal rather than to the ambient dimension of the noise vector. We connect our results to the existing literature and provide a discussion on their relation to the standard least-squares problem. Our error bounds are non-asymptotic and sharp, they apply to arbitrary convex functions and do not assume any distribution on the noise.