Hypothesis testing by convex optimization

Goldenshluger, Alexander; Juditsky, Anatoli; Nemirovski, Arkadi

doi:10.1214/15-ejs1054

Cited by 26 publications

(69 citation statements)

References 46 publications

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“…As shown in [15] (and can be immediately verified), the following o.s. 's are good: [25], Large Binocular Telescope [4,3], and Nanoscale Fluorescent Microscopy, a.k.a.…”

Section: Examples Of Good Observation Schemessupporting

confidence: 58%

“…Results of a typical experiment are presented in Figure 1. What follows is a summary of results of [15] which are relevant to our current needs. Assume that ω K = (ω 1 , ..., ω K ) is a stationary K-repeated observation in a good o.s.…”

Section: Illustrationmentioning

confidence: 99%

“…Let ∈ (0, 1/2), and suppose that there exists a test which, using a stationary K-repeated observation, decides on the hypotheses H 1 , H 2 with risk ≤ . Then 15) and the test T K with…”

Section: Illustrationmentioning

confidence: 99%

“…All we know in advance is that those risks are nearly as low as they can be under the circumstances.The main body of the paper is organized as follows. Section 2 contains preliminaries, originating from [21,15], on good o.s.'s. In Section 3 we deal with recovery of linear functions on the unions of convex sets.…”

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confidence: 99%

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Near-optimal recovery of linear and N-convex functions on unions of convex sets

Juditsky

Nemirovski

2019

Information and Inference: A Journal of the IMA

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In this paper we build provably near-optimal, in the minimax sense, estimates of linear forms and, more generally, "N -convex functionals" (an example being the maximum of several fractionallinear functions) of unknown "signal" from indirect noisy observations, the signal assumed to belong to the union of finitely many given convex compact sets. Our main assumption is that the observation scheme in question is good in the sense of [15], the simplest example being the Gaussian scheme where the observation is the sum of linear image of the signal and the standard Gaussian noise. The proposed estimates, same as upper bounds on their worst-case risks, stem from solutions to explicit convex optimization problems, making the estimates "computation-friendly." Immediate examples are affine-fractional functions f (x) = (a T x + a)/(b T x + b) with denominators positive on X , in particular, affine functions (N = 1), and piecewise linear functions like max[a T x + a, min[b T x + b, c T x + c]] (N = 3). A less trivial example is conditional quantile of a discrete distribution (N = 2), see Section 4.2.2 Our main results can be easily extended to the more general case of simple families -families of distributions specified in terms of upper bounds on their moment-generating functions, see [23,22] for details. Restricting the framework to the case of good observation schemes is aimed at streamlining the presentation.• Discrete o.s., where ω t are independent across t realizations of discrete random variable taking values 1, ..., m with probabilities affinely parameterized by x.The problem of (near-)optimal recovery of linear function f (x) on a convex compact set or a finite union of convex sets X has received much attention in the statistical literature (see, e.g., [20,10,12,13,14,11,7,8,9,21]). In particular, D. Donoho proved, see [11], that in the case of Gaussian observation scheme (1) and convex and compact X, the worst-case, over x ∈ X, risk of the minimax optimal affine in observations estimate is within factor 1.2 of the actual minimax risk. 3 Later, in [21], this near-optimality result was extended to other good observation schemes. In [8,9] the minimax affine estimator was used as "working horse" to build the near-optimal estimator of a linear functional over a finite union X of convex compact sets in the Gaussian observation scheme. As compared to the existing results, our contribution here is twofold. First, we pass from Gaussian o.s. to essentially more general good o.s.'s, extending in this respect the results of [8,9]. Second, we relax the requirement of affinity of the function to be recovered to N -convexity of the function. It should be stressed that the actual "common denominator" of the cited contributions and of the present work is the "operational nature" of the results, as opposed to typical results of non-parametric statistics which can be considered as descriptive. The traditional results present near-optimal estimates and their risks in a "closed analytical form," the toll being severe restrictions on...

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“…As shown in [15] (and can be immediately verified), the following o.s. 's are good: [25], Large Binocular Telescope [4,3], and Nanoscale Fluorescent Microscopy, a.k.a.…”

Section: Examples Of Good Observation Schemessupporting

confidence: 58%

Section: Illustrationmentioning

confidence: 99%

“…Let ∈ (0, 1/2), and suppose that there exists a test which, using a stationary K-repeated observation, decides on the hypotheses H 1 , H 2 with risk ≤ . Then 15) and the test T K with…”

Section: Illustrationmentioning

confidence: 99%

mentioning

confidence: 99%

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Near-optimal recovery of linear and N-convex functions on unions of convex sets

Juditsky

Nemirovski

2019

Information and Inference: A Journal of the IMA

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show abstract

“…In the situation considered in this paper, where we do not assume any specific structure of X apart from convexity, compactness and "computationally tractability," 1 and A, B are "general" matrices of appropriate dimensions, we cannot expect deriving closed form expressions for estimates and risks. Instead, we adopt an alternative approach initiated in [7] and further developed in [23,17,25,24]. Within this operational approach both the estimate and its risk are yielded by efficient computation, usually via convex optimization, rather than by an explicit closed form analytical description; what we know in advance, in good cases, is that the resulting risk, whether large or low, is nearly the best one achievable under the circumstances.…”

Section: Motivationmentioning

confidence: 99%

On polyhedral estimation of signals via indirect observations

Juditsky¹,

Nemirovski²

2020

Electron. J. Statist.

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We consider the problem of recovering linear image of unknown signal belonging to a given convex compact signal set from noisy observation of another linear image of the signal. We develop a simple generic efficiently computable nonlinear in observations "polyhedral" estimate along with computation-friendly techniques for its design and risk analysis. We demonstrate that under favorable circumstances the resulting estimate is provably nearoptimal in the minimax sense, the "favorable circumstances" being less restrictive than the weakest known so far assumptions ensuring near-optimality of estimates which are linear in observations.

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Constructive minimax classification of discrete observations with arbitrary loss function

Fillatre

2017

Signal Processing

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This paper develops a multihypothesis testing framework for calculating numerically the optimal minimax test with discrete observations and an arbitrary loss function. Discrete observations are common in data processing and make tractable the calculation of the minimax test. Each hypothesis is both associated to a parameter defining the distribution of the observations and to an action which describes the decision to take when the hypothesis is true. The loss function measures the gap between the parameters and the actions. The minimax test minimizes the maximum classification risk. It is the solution of a finite linear programming problem which gives the worst case classification risk and the worst case prior distribution. The minimax test equalizes the classification risks whose prior probabilities are strictly positive. The minimax framework is applied to vector channel decoding which consists in classifying some codewords transmitted on a binary asymmetric channel. The Hamming metric is used to measure the number of differences between the emitted codeword and the decoded one.

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Hypothesis testing by convex optimization

Cited by 26 publications

References 46 publications

Near-optimal recovery of linear and N-convex functions on unions of convex sets

Near-optimal recovery of linear and N-convex functions on unions of convex sets

On polyhedral estimation of signals via indirect observations

Constructive minimax classification of discrete observations with arbitrary loss function

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