The privacy-utility tradeoff problem is formulated as determining the privacy mechanism (random mapping) that minimizes the mutual information (a metric for privacy leakage) between the private features of the original dataset and a released version. The minimization is studied with two types of constraints on the distortion between the public features and the released version of the dataset: (i) subject to a constraint on the expected value of a cost function @math applied to the distortion, and (ii) subject to bounding the complementary CDF of the distortion by a non-increasing function @math . The first scenario captures various practical cost functions for distorted released data, while the second scenario covers large deviation constraints on utility. The asymptotic optimal leakage is derived in both scenarios. For the distortion cost constraint, it is shown that for convex cost functions there is no asymptotic loss in using stationary memoryless mechanisms. For the complementary CDF bound on distortion, the asymptotic leakage is derived for general mechanisms and shown to be the integral of the single letter leakage function with respect to the Lebesgue measure defined based on the refined bound on distortion. However, it is shown that memoryless mechanisms are generally suboptimal in both cases.
An alternative approach to more general distortion constraints is considered in @cite_8 and referred to as footnote 0 We have changed their notation from @math -separable to @math -separable, in order to avoid confusion with our notation. . In @cite_8 , a multi-letter distortion measure @math is defined as @math -separable if for an increasing function @math . The distortion cost constraints that we consider are more general in the sense that our notion of cost function @math applied to the distortion measure @math covers a broader class of distortion constraints than an average bound on @math -separable distortion measures studied in @cite_8 . Specifically, the average constraint on an @math -separable distortion measure has the form which clearly is a specific case for our formulation in that results from choosing @math and @math , such that @math . Moreover, we allow for non-decreasing functions @math , which means that @math does not have to be strictly increasing. We also note that our focus is on privacy rather than source coding.
@cite_8
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In this work we relax the usual separability assumption made in rate-distortion literature and propose f -separable distortion measures, which are well suited to model non-linear penalties. The main insight behind f -separable distortion measures is to define an n-letter distortion measure to be an f -mean of single-letter distortions. We prove a rate-distortion coding theorem for stationary ergodic sources with f -separable distortion measures, and provide some illustrative examples of the resulting rate-distortion functions. Finally, we discuss connections between f -separable distortion measures, and the subadditive distortion measure previously proposed in literature. Given a set of @math points in @math , how many dimensions are needed to represent all pair wise distances within a specific distortion? This dimension-distortion tradeoff question is well understood for the @math norm, where @math dimensions suffice to achieve @math distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in @math . A recent result shows that distortion @math can be achieved with @math dimensions. On the other hand, the only lower bounds known are that distortion @math requires @math dimensions and that distortion @math requires @math dimensions. In this work, we show the first near linear lower bounds for dimension reduction in @math . In particular, we show that @math distortion requires at least @math dimensions. Our proofs are combinatorial, but inspired by linear programming. In fact, our techniques lead to a simple combinatorial argument that is equivalent to the LP based proof of Brinkman-Charikar for lower bounds on dimension reduction in @math . Consider the recovery of an unknown signal @math from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that @math is sparse, and that the measurements are of the form @math . Since such measurements give no information on the norm of @math , recovery methods typically assume that @math . We show that if one allows more generally for quantized affine measurements of the form @math , and if the vectors @math are random, an appropriate choice of the affine shifts @math allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. In addition, we show that for arbitrary fixed @math in the annulus @math , one may estimate the norm @math up to additive error @math from @math such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors @math in a Euclidean ball of known radius. We investigate nonparametric multiproduct pricing problems, in which we want to find revenue maximizing prices for products @math based on a set of customer samples @math . We mostly focus on the unit-demand case, in which products constitute strict substitutes and each customer aims to purchase a single product. In this setting a customer sample consists of a number of nonzero values for different products and possibly an additional product ranking. Once prices are fixed, each customer chooses to buy one of the products she can afford based on some predefined selection rule. We distinguish between the min-buying, max-buying, and rank-buying models. Some of our results also extend to single-minded pricing, in which case products are strict complements and every customer seeks to buy a single set of products, which she purchases if the sum of prices is below her valuation for that set. For the min-buying model we show that the revenue maximization problem is not approximable within factor @math for some constant @math , unless @math , thereby almost closing the gap between the known algorithmic results and previous lower bounds. We also prove inapproximability within @math , @math being an upper bound on the number of nonzero values per customer, and @math under slightly stronger assumptions and provide matching upper bounds. Surprisingly, these hardness results hold even if a price ladder constraint, i.e., a predefined order on the prices of all products, is given. Without the price ladder constraint we obtain similar hardness results for the special case of uniform valuations, i.e., the case that every customer has identical values for all the products she is interested in, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas. Introducing a slightly more general problem definition in which customers are given as an explicit probability distribution, we obtain inapproximability within @math assuming @math . These results apply to single-minded pricing as well. For the max-buying model a polynomial-time approximation scheme exists if a price ladder is given. We give a matching lower bound by proving strong NP-hardness. Assuming limited product supply, we analyze a generic local search algorithm and prove that it is 2-approximate. Finally, we discuss implications for the rank-buying model.