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We consider the parameter estimation problem in the deviated Gaussian mixture of experts in which the data are generated from $(1 - \lambda^{\ast}) g_0(Y| X)+ \lambda^{\ast} \sum_{i = 1}^{k_{\ast}} p_{i}^{\ast} f(Y|(a_{i}^{\ast})^{\top}X+b_i^{\ast},\sigma_{i}^{\ast})$, where $X, Y$ are respectively a covariate vector and a response variable, $g_{0}(Y|X)$ is a known function, $\lambda^{\ast} \in [0, 1]$ is true but unknown mixing proportion, and $(p_{i}^{\ast}, a_{i}^{\ast}, b_{i}^{\ast}, \sigma_{i}^{\ast})$ for $1 \leq i \leq k^{\ast}$ are unknown parameters of the Gaussian mixture of experts. This problem arises from the goodness-of-fit test when we would like to test whether the data are generated from $g_{0}(Y|X)$ (null hypothesis) or they are generated from the whole mixture (alternative hypothesis). Based on the algebraic structure of the expert functions and the distinguishability between $g_0$ and the mixture part, we construct novel Voronoi-based loss functions to capture the convergence rates of maximum likelihood estimation (MLE) for our models. We further demonstrate that our proposed loss functions characterize the local convergence rates of parameter estimation more accurately than the generalized Wasserstein, a loss function being commonly used for estimating parameters in the Gaussian mixture of experts.

A randomized algorithm for computing a data sparse representation of a given rank structured matrix $A$ (a.k.a. an $H$-matrix) is presented. The algorithm draws on the randomized singular value decomposition (RSVD), and operates under the assumption that algorithms for rapidly applying $A$ and $A^{*}$ to vectors are available. The algorithm analyzes the hierarchical tree that defines the rank structure using graph coloring algorithms to generate a set of random test vectors. The matrix is then applied to the test vectors, and in a final step the matrix itself is reconstructed by the observed input-output pairs. The method presented is an evolution of the "peeling algorithm" of L. Lin, J. Lu, and L. Ying, "Fast construction of hierarchical matrix representation from matrix-vector multiplication," JCP, 230(10), 2011. For the case of uniform trees, the new method substantially reduces the pre-factor of the original peeling algorithm. More significantly, the new technique leads to dramatic acceleration for many non-uniform trees since it constructs sample vectors that are optimized for a given tree. The algorithm is particularly effective for kernel matrices involving a set of points restricted to a lower dimensional object than the ambient space, such as a boundary integral equation defined on a surface in three dimensions.

The $k$-CombDMR problem is that of determining whether an $n \times n$ distance matrix can be realised by $n$ vertices in some undirected graph with $n + k$ vertices. This problem has a simple solution in the case $k=0$. In this paper we show that this problem is polynomial time solvable for $k=1$ and $k=2$. Moreover, we provide algorithms to construct such graph realisations by solving appropriate 2-SAT instances. In the case where $k \geq 3$, this problem is NP-complete. We show this by a reduction of the $k$-colourability problem to the $k$-CombDMR problem. Finally, we discuss the simpler polynomial time solvable problem of tree realisability for a given distance matrix.

A key challenge of quantum programming is uncomputation: the reversible deallocation of qubits. And while there has been much recent progress on automating uncomputation, state-of-the-art methods are insufficient for handling today's expressive quantum programming languages. A core reason is that they operate on primitive quantum circuits, while quantum programs express computations beyond circuits, for instance, they can capture families of circuits defined recursively in terms of uncomputation and adjoints. In this paper, we introduce the first modular automatic approach to synthesize correct and efficient uncomputation for expressive quantum programs. Our method is based on two core technical contributions: (i) an intermediate representation (IR) that can capture expressive quantum programs and comes with support for uncomputation, and (ii) modular algorithms over that IR for synthesizing uncomputation and adjoints. We have built a complete end-to-end implementation of our method, including an implementation of the IR and the synthesis algorithms, as well as a translation from an expressive fragment of the Silq programming language to our IR and circuit generation from the IR. Our experimental evaluation demonstrates that we can handle programs beyond the capabilities of existing uncomputation approaches, while being competitive on the benchmarks they can handle. More broadly, we show that it is possible to benefit from the greater expressivity and safety offered by high-level quantum languages without sacrificing efficiency.

In the Levenshtein's sequence reconstruction problem a codeword is transmitted through $N$ channels and in each channel a set of errors is introduced to the transmitted word. In previous works, the restriction that each channel provides a unique output word has been essential. In this work, we assume only that each channel introduces a unique set of errors to the transmitted word and hence some output words can also be identical. As we will discuss, this interpretation is both natural and useful for deletion and insertion errors. We give properties, techniques and (optimal) results for this situation. Quaternary alphabets are relevant due to applications related to DNA-memories. Hence, we introduce an efficient Las Vegas style decoding algorithm for simultaneous insertion, deletion and substitution errors in $q$-ary Hamming spaces for $q \geq 4$.

Because $\Sigma^p_2$- and $\Sigma^p_3$-hardness proofs are usually tedious and difficult, not so many complete problems for these classes are known. This is especially true in the areas of min-max regret robust optimization, network interdiction, most vital vertex problems, blocker problems, and two-stage adjustable robust optimization problems. Even though these areas are well-researched for over two decades and one would naturally expect many (if not most) of the problems occurring in these areas to be complete for the above classes, almost no completeness results exist in the literature. We address this lack of knowledge by introducing over 70 new $\Sigma^p_2$-complete and $\Sigma^p_3$-complete problems. We achieve this result by proving a new meta-theorem, which shows $\Sigma^p_2$- and $\Sigma^p_3$-completeness simultaneously for a huge class of problems. The majority of all earlier publications on $\Sigma^p_2$- and $\Sigma^p_3$-completeness in said areas are special cases of our meta-theorem. Our precise result is the following: We introduce a large list of problems for which the meta-theorem is applicable (including clique, vertex cover, knapsack, TSP, facility location and many more). For every problem on this list, we show: The interdiction/minimum cost blocker/most vital nodes problem (with element costs) is $\Sigma^p_2$-complete. The min-max-regret problem with interval uncertainty is $\Sigma^p_2$-complete. The two-stage adjustable robust optimization problem with discrete budgeted uncertainty is $\Sigma^p_3$-complete. In summary, our work reveals the interesting insight that a large amount of NP-complete problems have the property that their min-max versions are 'automatically' $\Sigma^p_2$-complete.

Model order reduction methods are a powerful tool to drastically reduce the computational effort of problems which need to be evaluated repeatedly, i.e., when computing the same system for various parameter values. When applying a reduced basis approximation algorithm to the Maxwell eigenvalue problem, we encounter spurious solutions in the reduced system which hence need to be removed during the basis construction. In this paper, we discuss two tree-cotree gauge-based methods for the removal of the spurious eigenmodes.

Given a finite family $\mathcal{F}$ of graphs, we say that a graph $G$ is "$\mathcal{F}$-free" if $G$ does not contain any graph in $\mathcal{F}$ as a subgraph. A vertex-colored graph $H$ is called "rainbow" if no two vertices of $H$ have the same color. Given an integer $s$ and a finite family of graphs $\mathcal{F}$, let $\ell(s,\mathcal{F})$ denote the smallest integer such that any properly vertex-colored $\mathcal{F}$-free graph $G$ having $\chi(G)\geq\ell(s,\mathcal{F})$ contains an induced rainbow path on $s$ vertices. Scott and Seymour showed that $\ell(s,K)$ exists for every complete graph $K$. A conjecture of N. R. Aravind states that $\ell(s,C_3)=s$. The upper bound on $\ell(s,C_3)$ that can be obtained using the methods of Scott and Seymour setting $K=C_3$ are, however, super-exponential. Gy\'arf\'as and S\'ark\"ozy showed that $\ell(s,\{C_3,C_4\})=\mathcal{O}\big((2s)^{2s}\big)$. For $r\geq 2$, we show that $\ell(s,K_{2,r})\leq (r-1)(s-1)(s-2)/2+s$ and therefore, $\ell(s,C_4)\leq\frac{s^2-s+2}{2}$. This significantly improves Gy\'arf\'as and S\'ark\"ozy's bound and also covers a bigger class of graphs. We adapt our proof to achieve much stronger upper bounds for graphs of higher girth: we prove that $\ell(s,\{C_3,C_4,\ldots,C_{g-1}\})\leq s^{1+\frac{4}{g-4}}$, where $g\geq 5$. Moreover, in each case, our results imply the existence of at least $s!/2$ distinct induced rainbow paths on $s$ vertices. Along the way, we obtain some results on related problems on oriented graphs. For $r\geq 2$, let $\mathcal{B}_r$ denote the orientations of $K_{2,r}$ in which one vertex has out-degree or in-degree $r$. We show that every $\mathcal{B}_r$-free oriented graph $G$ having $\chi(G)\geq (r-1)(s-1)(s-2)+2s+1$ and every bikernel-perfect oriented graph $G$ with girth $g\geq 5$ having $\chi(G)\geq 2s^{1+\frac{4}{g-4}}$ contains every $s$ vertex oriented tree as an induced subgraph.

A toric code, introduced by Hansen to extend the Reed-Solomon code as a $k$-dimensional subspace of $\mathbb{F}_q^n$, is determined by a toric variety or its associated integral convex polytope $P \subseteq [0,q-2]^n$, where $k=|P \cap \mathbb{Z}^n|$ (the number of integer lattice points of $P$). There are two relevant parameters that determine the quality of a code: the information rate, which measures how much information is contained in a single bit of each codeword; and the relative minimum distance, which measures how many errors can be corrected relative to how many bits each codeword has. Soprunov and Soprunova defined a good infinite family of codes to be a sequence of codes of unbounded polytope dimension such that neither the corresponding information rates nor relative minimum distances go to 0 in the limit. We examine different ways of constructing families of codes by considering polytope operations such as the join and direct sum. In doing so, we give conditions under which no good family can exist and strong evidence that there is no such good family of codes.

Object detection typically assumes that training and test data are drawn from an identical distribution, which, however, does not always hold in practice. Such a distribution mismatch will lead to a significant performance drop. In this work, we aim to improve the cross-domain robustness of object detection. We tackle the domain shift on two levels: 1) the image-level shift, such as image style, illumination, etc, and 2) the instance-level shift, such as object appearance, size, etc. We build our approach based on the recent state-of-the-art Faster R-CNN model, and design two domain adaptation components, on image level and instance level, to reduce the domain discrepancy. The two domain adaptation components are based on H-divergence theory, and are implemented by learning a domain classifier in adversarial training manner. The domain classifiers on different levels are further reinforced with a consistency regularization to learn a domain-invariant region proposal network (RPN) in the Faster R-CNN model. We evaluate our newly proposed approach using multiple datasets including Cityscapes, KITTI, SIM10K, etc. The results demonstrate the effectiveness of our proposed approach for robust object detection in various domain shift scenarios.