TY - CONF AB - Fejes TΓ³th [5] and Schneider [9] studied approximations of smooth convex hypersurfaces in Euclidean space by piecewise flat triangular meshes with a given number of vertices on the hypersurface that are optimal with respect to Hausdorff distance. They proved that this Hausdorff distance decreases inversely proportional with m 2/(dβˆ’1), where m is the number of vertices and d is the dimension of Euclidean space. Moreover the pro-portionality constant can be expressed in terms of the Gaussian curvature, an intrinsic quantity. In this short note, we prove the extrinsic nature of this constant for manifolds of sufficiently high codimension. We do so by constructing an family of isometric embeddings of the flat torus in Euclidean space. AU - Vegter, Gert AU - Wintraecken, Mathijs ID - 6628 T2 - The 31st Canadian Conference in Computational Geometry TI - The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds ER - TY - CONF AB - Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context. AU - Edelsbrunner, Herbert AU - Virk, Ziga AU - Wagner, Hubert ID - 6648 SN - 9783959771047 T2 - 35th International Symposium on Computational Geometry TI - Topological data analysis in information space VL - 129 ER - TY - JOUR AB - Chemical labeling of proteins with synthetic molecular probes offers the possibility to probe the functions of proteins of interest in living cells. However, the methods for covalently labeling targeted proteins using complementary peptide tag-probe pairs are still limited, irrespective of the versatility of such pairs in biological research. Herein, we report the new CysHis tag-Ni(II) probe pair for the specific covalent labeling of proteins. A broad-range evaluation of the reactivity profiles of the probe and the CysHis peptide tag afforded a tag-probe pair with an optimized and high labeling selectivity and reactivity. In particular, the labeling specificity of this pair was notably improved compared to the previously reported one. This pair was successfully utilized for the fluorescence imaging of membrane proteins on the surfaces of living cells, demonstrating its potential utility in biological research. AU - Zenmyo, Naoki AU - Tokumaru, Hiroki AU - Uchinomiya, Shohei AU - Fuchida, Hirokazu AU - Tabata, Shigekazu AU - Hamachi, Itaru AU - Shigemoto, Ryuichi AU - Ojida, Akio ID - 6659 IS - 5 JF - Bulletin of the Chemical Society of Japan SN - 00092673 TI - Optimized reaction pair of the CysHis tag and Ni(II)-NTA probe for highly selective chemical labeling of membrane proteins VL - 92 ER - TY - JOUR AB - In phase retrieval, we want to recover an unknown signal π‘₯βˆˆβ„‚π‘‘ from n quadratic measurements of the form 𝑦𝑖=|βŸ¨π‘Žπ‘–,π‘₯⟩|2+𝑀𝑖, where π‘Žπ‘–βˆˆβ„‚π‘‘ are known sensing vectors and 𝑀𝑖 is measurement noise. We ask the following weak recovery question: What is the minimum number of measurements n needed to produce an estimator π‘₯^(𝑦) that is positively correlated with the signal π‘₯? We consider the case of Gaussian vectors π‘Žπ‘Žπ‘–. We prove thatβ€”in the high-dimensional limitβ€”a sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For π‘›β‰€π‘‘βˆ’π‘œ(𝑑), no estimator can do significantly better than random and achieve a strictly positive correlation. For 𝑛β‰₯𝑑+π‘œ(𝑑), a simple spectral estimator achieves a positive correlation. Surprisingly, numerical simulations with the same spectral estimator demonstrate promising performance with realistic sensing matrices. Spectral methods are used to initialize non-convex optimization algorithms in phase retrieval, and our approach can boost the performance in this setting as well. Our impossibility result is based on classical information-theoretic arguments. The spectral algorithm computes the leading eigenvector of a weighted empirical covariance matrix. We obtain a sharp characterization of the spectral properties of this random matrix using tools from free probability and generalizing a recent result by Lu and Li. Both the upper bound and lower bound generalize beyond phase retrieval to measurements 𝑦𝑖 produced according to a generalized linear model. As a by-product of our analysis, we compare the threshold of the proposed spectral method with that of a message passing algorithm. AU - Mondelli, Marco AU - Montanari, Andrea ID - 6662 IS - 3 JF - Foundations of Computational Mathematics TI - Fundamental limits of weak recovery with applications to phase retrieval VL - 19 ER - TY - JOUR AB - The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in $\mathbb{R}^2$ and on surfaces embedded in $\mathbb{R}^3$ as detailed in our experimental companion paper. In this paper, we study theoretical aspects of our structure. Given a finite set of points $\mathcal{P}$ in a domain $\Omega$ equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of $\mathcal{P}$ to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in $\Omega$ under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened. AU - Boissonnat, Jean-Daniel AU - Rouxel-LabbΓ©, Mael AU - Wintraecken, Mathijs ID - 6672 IS - 3 JF - SIAM Journal on Computing SN - 0097-5397 TI - Anisotropic triangulations via discrete Riemannian Voronoi diagrams VL - 48 ER - TY - CONF AB - A Valued Constraint Satisfaction Problem (VCSP) provides a common framework that can express a wide range of discrete optimization problems. A VCSP instance is given by a finite set of variables, a finite domain of labels, and an objective function to be minimized. This function is represented as a sum of terms where each term depends on a subset of the variables. To obtain different classes of optimization problems, one can restrict all terms to come from a fixed set Ξ“ of cost functions, called a language. Recent breakthrough results have established a complete complexity classification of such classes with respect to language Ξ“: if all cost functions in Ξ“ satisfy a certain algebraic condition then all Ξ“-instances can be solved in polynomial time, otherwise the problem is NP-hard. Unfortunately, testing this condition for a given language Ξ“ is known to be NP-hard. We thus study exponential algorithms for this meta-problem. We show that the tractability condition of a finite-valued language Ξ“ can be tested in O(3β€Ύβˆš3|D|β‹…poly(size(Ξ“))) time, where D is the domain of Ξ“ and poly(β‹…) is some fixed polynomial. We also obtain a matching lower bound under the Strong Exponential Time Hypothesis (SETH). More precisely, we prove that for any constant Ξ΄<1 there is no O(3β€Ύβˆš3Ξ΄|D|) algorithm, assuming that SETH holds. AU - Kolmogorov, Vladimir ID - 6725 SN - 1868-8969 T2 - 46th International Colloquium on Automata, Languages and Programming TI - Testing the complexity of a valued CSP language VL - 132 ER - TY - CHAP AB - Randomness is an essential part of any secure cryptosystem, but many constructions rely on distributions that are not uniform. This is particularly true for lattice based cryptosystems, which more often than not make use of discrete Gaussian distributions over the integers. For practical purposes it is crucial to evaluate the impact that approximation errors have on the security of a scheme to provide the best possible trade-off between security and performance. Recent years have seen surprising results allowing to use relatively low precision while maintaining high levels of security. A key insight in these results is that sampling a distribution with low relative error can provide very strong security guarantees. Since floating point numbers provide guarantees on the relative approximation error, they seem a suitable tool in this setting, but it is not obvious which sampling algorithms can actually profit from them. While previous works have shown that inversion sampling can be adapted to provide a low relative error (PΓΆppelmann et al., CHES 2014; Prest, ASIACRYPT 2017), other works have called into question if this is possible for other sampling techniques (Zheng et al., Eprint report 2018/309). In this work, we consider all sampling algorithms that are popular in the cryptographic setting and analyze the relationship of floating point precision and the resulting relative error. We show that all of the algorithms either natively achieve a low relative error or can be adapted to do so. AU - Walter, Michael ED - Buchmann, J ED - Nitaj, A ED - Rachidi, T ID - 6726 SN - 0302-9743 T2 - Progress in Cryptology – AFRICACRYPT 2019 TI - Sampling the integers with low relative error VL - 11627 ER - TY - JOUR AB - Consider the problem of constructing a polar code of block length N for a given transmission channel W. Previous approaches require one to compute the reliability of the N synthetic channels and then use only those that are sufficiently reliable. However, we know from two independent works by SchΓΌrch and by Bardet et al. that the synthetic channels are partially ordered with respect to degradation. Hence, it is natural to ask whether the partial order can be exploited to reduce the computational burden of the construction problem. We show that, if we take advantage of the partial order, we can construct a polar code by computing the reliability of roughly a fraction 1/ log 3/2 N of the synthetic channels. In particular, we prove that N/ log 3/2 N is a lower bound on the number of synthetic channels to be considered and such a bound is tight up to a multiplicative factor log log N. This set of roughly N/ log 3/2 N synthetic channels is universal, in the sense that it allows one to construct polar codes for any W, and it can be identified by solving a maximum matching problem on a bipartite graph. Our proof technique consists of reducing the construction problem to the problem of computing the maximum cardinality of an antichain for a suitable partially ordered set. As such, this method is general, and it can be used to further improve the complexity of the construction problem, in case a refined partial order on the synthetic channels of polar codes is discovered. AU - Mondelli, Marco AU - Hassani, Hamed AU - Urbanke, Rudiger ID - 6663 IS - 5 JF - IEEE TI - Construction of polar codes with sublinear complexity VL - 65 ER - TY - CONF AB - We establish connections between the problem of learning a two-layer neural network and tensor decomposition. We consider a model with feature vectors xβˆˆβ„d, r hidden units with weights {wi}1≀i≀r and output yβˆˆβ„, i.e., y=βˆ‘ri=1Οƒ(w𝖳ix), with activation functions given by low-degree polynomials. In particular, if Οƒ(x)=a0+a1x+a3x3, we prove that no polynomial-time learning algorithm can outperform the trivial predictor that assigns to each example the response variable 𝔼(y), when d3/2β‰ͺrβ‰ͺd2. Our conclusion holds for a `natural data distribution', namely standard Gaussian feature vectors x, and output distributed according to a two-layer neural network with random isotropic weights, and under a certain complexity-theoretic assumption on tensor decomposition. Roughly speaking, we assume that no polynomial-time algorithm can substantially outperform current methods for tensor decomposition based on the sum-of-squares hierarchy. We also prove generalizations of this statement for higher degree polynomial activations, and non-random weight vectors. Remarkably, several existing algorithms for learning two-layer networks with rigorous guarantees are based on tensor decomposition. Our results support the idea that this is indeed the core computational difficulty in learning such networks, under the stated generative model for the data. As a side result, we show that under this model learning the network requires accurate learning of its weights, a property that does not hold in a more general setting. AU - Mondelli, Marco AU - Montanari, Andrea ID - 6747 T2 - Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics TI - On the connection between learning two-layers neural networks and tensor decomposition VL - 89 ER - TY - JOUR AB - Polar codes have gained extensive attention during the past few years and recently they have been selected for the next generation of wireless communications standards (5G). Successive-cancellation-based (SC-based) decoders, such as SC list (SCL) and SC flip (SCF), provide a reasonable error performance for polar codes at the cost of low decoding speed. Fast SC-based decoders, such as Fast-SSC, Fast-SSCL, and Fast-SSCF, identify the special constituent codes in a polar code graph off-line, produce a list of operations, store the list in memory, and feed the list to the decoder to decode the constituent codes in order efficiently, thus increasing the decoding speed. However, the list of operations is dependent on the code rate and as the rate changes, a new list is produced, making fast SC-based decoders not rate-flexible. In this paper, we propose a completely rate-flexible fast SC-based decoder by creating the list of operations directly in hardware, with low implementation complexity. We further propose a hardware architecture implementing the proposed method and show that the area occupation of the rate-flexible fast SC-based decoder in this paper is only 38% of the total area of the memory-based base-line decoder when 5G code rates are supported. AU - Hashemi, Seyyed Ali AU - Condo, Carlo AU - Mondelli, Marco AU - Gross, Warren J ID - 6750 IS - 22 JF - IEEE Transactions on Signal Processing SN - 1053587X TI - Rate-flexible fast polar decoders VL - 67 ER -