This paper is concerned with the robustness of VAEs to adversarial attacks. We highlight that conventional VAEs are brittle under attack but that methods recently introduced for disentanglement such as @math -TCVAE (, 2018) improve robustness, as demonstrated through a variety of previously proposed adversarial attacks ( (2016); Gondim- (2018); (2018)). This motivated us to develop Seatbelt-VAE, a new hierarchical disentangled VAE that is designed to be significantly more robust to adversarial attacks than existing approaches, while retaining high quality reconstructions.
In a @math -VAE @cite_14 , a free parameter @math multiplies the @math term in @math above. This objective @math remains a lower bound on the evidence.
@cite_14
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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. We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze and generalize the Belief Propagation (BP) approach to computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. We introduce a fractional free energy functional parameterized by a scalar parameter @math , where @math corresponds to the BP limit and @math corresponds to the exclusion principle Mean-Field (MF) limit, and show monotonicity and continuity of the functional with @math . We observe that the optimal value of @math , where the @math -parameterized functional is equal to the exact free energy (defined as the minus log of the permanent), lies in the @math range, with the low and high values from the range producing provable low and upper bounds for the permanent. Our experimental analysis suggests that the optimal @math varies for different ensembles considered but it always lies in the @math interval. Besides, for all ensembles considered the behavior of the optimal @math is highly distinctive, thus offering a lot of practical potential for estimating permanents of non-negative matrices via the fractional free energy functional approach. We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is @math -complete to decide whether the classical value of a non-local game is 1 or @math . Furthermore, as long as @math is small enough, this result does not depend on the gap @math . In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap @math decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of @math (proofs with two provers, one round, completeness probability @math , soundness probability @math , and commuting-operator strategies) can increase without bound as the gap @math gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class @math - @math of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap @math gets arbitrarily small. While we do not know any computable time upper bound on the class @math , a result of the first author and Vidick shows that for @math and @math , the class @math , with constant communication from the provers, is contained in @math . We give a lower bound of @math (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis. We decompose the evidence lower bound to show the existence of a term measuring the total correlation between latent variables. We use this to motivate our @math -TCVAE (Total Correlation Variational Autoencoder), a refinement of the state-of-the-art @math -VAE objective for learning disentangled representations, requiring no additional hyperparameters during training. We further propose a principled classifier-free measure of disentanglement called the mutual information gap (MIG). We perform extensive quantitative and qualitative experiments, in both restricted and non-restricted settings, and show a strong relation between total correlation and disentanglement, when the latent variables model is trained using our framework.