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 -