We propose a new variant of Chubanov's method for solving the feasibility problem over the symmetric cone by extending Roos's method (2018) for the feasibility problem over the nonnegative orthant. The proposed method considers a feasibility problem associated with a norm induced by the maximum eigenvalue of an element and uses a rescaling focusing on the upper bound of the sum of eigenvalues of any feasible solution to the problem. Its computational bound is (i) equivalent to Roos's original method (2018) and superior to Louren\c{c}o et al.'s method (2019) when the symmetric cone is the nonnegative orthant, (ii) superior to Louren\c{c}o et al.'s method (2019) when the symmetric cone is a Cartesian product of second-order cones, and (iii) equivalent to Louren\c{c}o et al.'s method (2019) when the symmetric cone is the simple positive semidefinite cone, under the assumption that the costs of computing the spectral decomposition and the minimum eigenvalue are of the same order for any given symmetric matrix. We also conduct numerical experiments that compare the performance of our method with existing methods by generating instance in three types: (i) strongly (but ill-conditioned) feasible instances, (ii) weakly feasible instances, and (iii) infeasible instances. For any of these instances, the proposed method is rather more efficient than the existing methods in terms of accuracy and execution time.

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There is a one-to-one correspondence between L\'{e}vy copulas and proper copulas. The correspondence relies on a relationship between L\'{e}vy copulas sitting on $[0,+\infty]^d$ and max-id distributions. The max-id distributions are defined with respect to a partial order that is compatible with the inclusion of sets bounded away from the origin. An important consequence of the result is the possibility to define parametric L\'{e}vy copulas as mirror images of proper parametric copulas. For example, proper Archimedean copulas are generated by functions that are Williamson $d-$transforms of the cdf of the radial component of random vectors with exchangeable distributions $F_{R}$. In contrast, the generators of Archimedean L\'{e}vy copulas are Williamson $d-$transforms of $-\log(1-F_{R})$.

In the Geometric Median problem with outliers, we are given a finite set of points in d-dimensional real space and an integer m, the goal is to locate a new point in space (center) and choose m of the input points to minimize the sum of the Euclidean distances from the center to the chosen points. This problem can be solved "almost exactly" in polynomial time if d is fixed and admits an approximation scheme PTAS in high dimensions. However, the complexity of the problem was an open question. We prove that, if the dimension of space is not fixed, Geometric Median with outliers is strongly NP-hard, does not admit approximation schemes FPTAS unless P=NP, and is W[1]-hard with respect to the parameter m. The proof is done by a reduction from the Independent Set problem. Based on a similar reduction, we also get the NP-hardness of closely related geometric 2-clustering problems in which it is required to partition a given set of points into two balanced clusters minimizing the cost of median clustering. Finally, we study Geometric Median with outliers in $\ell_\infty$ space and prove the same complexity results as for the Euclidean problem.

We deal with a long-standing problem about how to design an energy-stable numerical scheme for solving the motion of a closed curve under {\sl anisotropic surface diffusion} with a general anisotropic surface energy $\gamma(\boldsymbol{n})$ in two dimensions, where $\boldsymbol{n}$ is the outward unit normal vector. By introducing a novel symmetric positive definite surface energy matrix $Z_k(\boldsymbol{n})$ depending on the Cahn-Hoffman $\boldsymbol{\xi}$-vector and a stabilizing function $k(\boldsymbol{n})$, we first reformulate the anisotropic surface diffusion into a conservative form and then derive a new symmetrized variational formulation for the anisotropic surface diffusion with both weakly and strongly anisotropic surface energies. A semi-discretization in space for the symmetrized variational formulation is proposed and its area (or mass) conservation and energy dissipation are proved. The semi-discretization is then discretized in time by either an implicit structural-preserving scheme (SP-PFEM) which preserves the area in the discretized level or a semi-implicit energy-stable method (ES-PFEM) which needs only solve a linear system at each time step. Under a relatively simple and mild condition on $\gamma(\boldsymbol{n})$, we show that both SP-PFEM and ES-PFEM are energy dissipative and thus are unconditionally energy-stable for almost all anisotropic surface energies $\gamma(\boldsymbol{n})$ arising in practical applications. Specifically, for several commonly-used anisotropic surface energies, we construct $Z_k(\boldsymbol{n})$ explicitly. Finally, extensive numerical results are reported to demonstrate the efficiency and accuracy as well as the unconditional energy-stability of the proposed symmetrized parametric finite element method.

We introduce a natural online allocation problem that connects several of the most fundamental problems in online optimization. Let $M$ be an $n$-point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of $M$. At each time $t$, a convex monotone cost function $c_t: [0,1]\to\mathbb{R}_+$ appears at some point $r_t\in M$. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost $c_t(x_{r_t})$, where $x_{r_t}$ is the fraction of the resource at $r_t$ at the end of time $t$. For example, when the cost functions are $c_t(x)=\alpha x$, this is equivalent to randomized MTS, and when the cost functions are $c_t(x)=\infty\cdot 1_{x<1/k}$, this is equivalent to fractional $k$-server. We give an $O(\log n)$-competitive algorithm for weighted star metrics. Due to the generality of allowed cost functions, classical multiplicative update algorithms do not work for the metric allocation problem. A key idea of our algorithm is to decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. This can be viewed as running mirror descent with a time-varying regularizer, and we use this perspective to further refine the guarantees of our algorithm. The standard analysis techniques run into multiple complications when the regularizer is time-varying, and we show how to overcome these issues by making various modifications to the default potential function. We also consider the problem when cost functions are allowed to be non-convex. In this case, we give tight bounds of $\Theta(n)$ on tree metrics, which imply deterministic and randomized competitive ratios of $O(n^2)$ and $O(n\log n)$ respectively on arbitrary metrics. Our algorithm is based on an $\ell_2^2$-regularizer.

This paper deals with the grouped variable selection problem. A widely used strategy is to equip the loss function with a sparsity-promoting penalty. Existing methods include the group Lasso, group SCAD, and group MCP. The group Lasso solves a convex optimization problem but is plagued by underestimation bias. The group SCAD and group MCP avoid the estimation bias but require solving a non-convex optimization problem that suffers from local optima. In this work, we propose an alternative method based on the generalized minimax concave (GMC) penalty, which is a folded concave penalty that can maintain the convexity of the objective function. We develop a new method for grouped variable selection in linear regression, the group GMC, that generalizes the strategy of the original GMC estimator. We present an efficient algorithm for computing the group GMC estimator. We also prove properties of the solution path to guide its numerical computation and tuning parameter selection in practice. We establish error bounds for both the group GMC and original GMC estimators. A rich set of simulation studies and a real data application indicate that the proposed group GMC approach outperforms existing methods in several different aspects under a wide array of scenarios.

In this paper, we introduce the online and streaming MAP inference and learning problems for Non-symmetric Determinantal Point Processes (NDPPs) where data points arrive in an arbitrary order and the algorithms are constrained to use a single-pass over the data as well as sub-linear memory. The online setting has an additional requirement of maintaining a valid solution at any point in time. For solving these new problems, we propose algorithms with theoretical guarantees, evaluate them on several real-world datasets, and show that they give comparable performance to state-of-the-art offline algorithms that store the entire data in memory and take multiple passes over it.

We revisit the satisfiability problem for two-variable logic, denoted by SAT(FO2), which is known to be NEXP-complete. The upper bound is usually derived from its well known Exponential Size Model (ESM) property. Whether it can be determinized efficiently is still an open question. In this paper we present a different approach by reducing it to a novel graph-theoretic problem that we call Conditional Independent Set (CIS). We show that CIS is NP-complete and present two simple algorithms for it with run time O(1.4423^n) and O(1.6181^n), where n is the number of vertices in the graph. We also show that unless the "Strong Exponential Time Hypothesis" (SETH) fails, there is no algorithm for CIS with run time O(1.4141^n). We show that without the equality predicate SAT(FO2) is in fact equivalent to CIS in succinct representation. This yields two algorithms for SAT(FO2) without the equality predicate with run time O(1.4423^{2^n}) and O(1.6181^{2^n}), where n is the number of predicates. To the best of our knowledge, these are the first exact algorithms for an NEXP-complete decidable logic with run time significantly lower than O(2^{2^n}). We also identify a few lower complexity fragments of FO2 which correspond to the tractable fragments of CIS. Similar to CIS, unless SETH fails, there is no algorithm for SAT(FO2) with run time O(1.4141^{2^n}). For the fragment with the equality predicate, we present a linear time many-one reduction to the fragment without the equality predicate. The reduction yields equi-satisfiable formulas with a small constant blow-up in the number of predicates. Finally, we also perform some small experiments which show that our approach is indeed more promising than the existing method (based on the ESM property). The experiments also show that although theoretically it has the worse run time, the second algorithm in general performs better than the first one.

We consider mixed finite element methods with exact symmetric stress tensors. We derive a new quasi-optimal a priori error estimate uniformly valid with respect to the compressibility. For the a posteriori error analysis we consider the Prager-Synge hypercircle principle and introduce a new estimate uniformly valid in the incompressible limit. All estimates are validated by numerical examples.

The tensor power method generalizes the matrix power method to higher order arrays, or tensors. Like in the matrix case, the fixed points of the tensor power method are the eigenvectors of the tensor. While every real symmetric matrix has an eigendecomposition, the vectors generating a symmetric decomposition of a real symmetric tensor are not always eigenvectors of the tensor. In this paper we show that whenever an eigenvector is a generator of the symmetric decomposition of a symmetric tensor, then (if the order of the tensor is sufficiently high) this eigenvector is robust, i.e., it is an attracting fixed point of the tensor power method. We exhibit new classes of symmetric tensors whose symmetric decomposition consists of eigenvectors. Generalizing orthogonally decomposable tensors, we consider equiangular tight frame decomposable and equiangular set decomposable tensors. Our main result implies that such tensors can be decomposed using the tensor power method.

The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fr\'echet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded.

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