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The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle-point formulation is well established since many decades. However, this topic was mostly studied for variational formulations defined upon the same finite-element product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever these two product spaces are different the saddle point problem is asymmetric. It turns out that the conditions to be satisfied by the finite elements product spaces stipulated in the few works on this case may be of limited use in practice. The purpose of this paper is to provide an in-depth analysis of the well-posedness and the uniform stability of asymmetric approximate saddle point problems, based on the theory of continuous linear operators on Hilbert spaces. Our approach leads to necessary and sufficient conditions for such properties to hold, expressed in a readily exploitable form with fine constants. In particular standard interpolation theory suffices to estimate the error of a conforming method.

Variational quantum algorithms (VQAs) prevail to solve practical problems such as combinatorial optimization, quantum chemistry simulation, quantum machine learning, and quantum error correction on noisy quantum computers. For variational quantum machine learning, a variational algorithm with model interpretability built into the algorithm is yet to be exploited. In this paper, we construct a quantum regression algorithm and identify the direct relation of variational parameters to learned regression coefficients, while employing a circuit that directly encodes the data in quantum amplitudes reflecting the structure of the classical data table. The algorithm is particularly suitable for well-connected qubits. With compressed encoding and digital-analog gate operation, the run time complexity is logarithmically more advantageous than that for digital 2-local gate native hardware with the number of data entries encoded, a decent improvement in noisy intermediate-scale quantum computers and a minor improvement for large-scale quantum computing Our suggested method of compressed binary encoding offers a remarkable reduction in the number of physical qubits needed when compared to the traditional one-hot-encoding technique with the same input data. The algorithm inherently performs linear regression but can also be used easily for nonlinear regression by building nonlinear features into the training data. In terms of measured cost function which distinguishes a good model from a poor one for model training, it will be effective only when the number of features is much less than the number of records for the encoded data structure to be observable. To echo this finding and mitigate hardware noise in practice, the ensemble model training from the quantum regression model learning with important feature selection from regularization is incorporated and illustrated numerically.

For quantum error-correcting codes to be realizable, it is important that the qubits subject to the code constraints exhibit some form of limited connectivity. The works of Bravyi & Terhal (BT) and Bravyi, Poulin & Terhal (BPT) established that geometric locality constrains code properties -- for instance $[[n,k,d]]$ quantum codes defined by local checks on the $D$-dimensional lattice must obey $k d^{2/(D-1)} \le O(n)$. Baspin and Krishna studied the more general question of how the connectivity graph associated with a quantum code constrains the code parameters. These trade-offs apply to a richer class of codes compared to the BPT and BT bounds, which only capture geometrically-local codes. We extend and improve this work, establishing a tighter dimension-distance trade-off as a function of the size of separators in the connectivity graph. We also obtain a distance bound that covers all stabilizer codes with a particular separation profile, rather than only LDPC codes.

Learning with rejection is a prototypical model for studying the interaction between humans and AI on prediction tasks. The model has two components, a predictor and a rejector. Upon the arrival of a sample, the rejector first decides whether to accept it; if accepted, the predictor fulfills the prediction task, and if rejected, the prediction will be deferred to humans. The learning problem requires learning a predictor and a rejector simultaneously. This changes the structure of the conventional loss function and often results in non-convexity and inconsistency issues. For the classification with rejection problem, several works develop surrogate losses for the jointly learning with provable consistency guarantees; in parallel, there has been less work for the regression counterpart. We study the regression with rejection (RwR) problem and investigate the no-rejection learning strategy which treats the RwR problem as a standard regression task to learn the predictor. We establish that the suboptimality of the no-rejection learning strategy observed in the literature can be mitigated by enlarging the function class of the predictor. Then we introduce the truncated loss to single out the learning for the predictor and we show that a consistent surrogate property can be established for the predictor individually in an easier way than for the predictor and the rejector jointly. Our findings advocate for a two-step learning procedure that first uses all the data to learn the predictor and then calibrates the prediction loss for the rejector. It is better aligned with the common intuition that more data samples will lead to a better predictor and it calls for more efforts on a better design of calibration algorithms for learning the rejector. While our discussions mainly focus on the regression problem, the theoretical results and insights generalize to the classification problem as well.

We propose a novel a-posteriori error estimation technique where the target quantities of interest are ratios of high-dimensional integrals, as occur e.g. in PDE constrained Bayesian inversion and PDE constrained optimal control subject to an entropic risk measure. We consider in particular parametric, elliptic PDEs with affine-parametric diffusion coefficient, on high-dimensional parameter spaces. We combine our recent a-posteriori Quasi-Monte Carlo (QMC) error analysis, with Finite Element a-posteriori error estimation. The proposed approach yields a computable a-posteriori estimator which is reliable, up to higher order terms. The estimator's reliability is uniform with respect to the PDE discretization, and robust with respect to the parametric dimension of the uncertain PDE input.

The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.

Time-series clustering serves as a powerful data mining technique for time-series data in the absence of prior knowledge about clusters. A large amount of time-series data with large size has been acquired and used in various research fields. Hence, clustering method with low computational cost is required. Given that a quantum-inspired computing technology, such as a simulated annealing machine, surpasses conventional computers in terms of fast and accurately solving combinatorial optimization problems, it holds promise for accomplishing clustering tasks that are challenging to achieve using existing methods. This study proposes a novel time-series clustering method that leverages an annealing machine. The proposed method facilitates an even classification of time-series data into clusters close to each other while maintaining robustness against outliers. Moreover, its applicability extends to time-series images. We compared the proposed method with a standard existing method for clustering an online distributed dataset. In the existing method, the distances between each data are calculated based on the Euclidean distance metric, and the clustering is performed using the k-means++ method. We found that both methods yielded comparable results. Furthermore, the proposed method was applied to a flow measurement image dataset containing noticeable noise with a signal-to-noise ratio of approximately 1. Despite a small signal variation of approximately 2%, the proposed method effectively classified the data without any overlap among the clusters. In contrast, the clustering results by the standard existing method and the conditional image sampling (CIS) method, a specialized technique for flow measurement data, displayed overlapping clusters. Consequently, the proposed method provides better results than the other two methods, demonstrating its potential as a superior clustering method.

Maximum weight independent set (MWIS) admits a $\frac1k$-approximation in inductively $k$-independent graphs and a $\frac{1}{2k}$-approximation in $k$-perfectly orientable graphs. These are a a parameterized class of graphs that generalize $k$-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others. We consider a generalization of MWIS to a submodular objective. Given a graph $G=(V,E)$ and a non-negative submodular function $f: 2^V \rightarrow \mathbb{R}_+$, the goal is to approximately solve $\max_{S \in \mathcal{I}_G} f(S)$ where $\mathcal{I}_G$ is the set of independent sets of $G$. We obtain an $\Omega(\frac1k)$-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least $\frac{1}{e(k+1)}$. This approach also yields parallel (or low-adaptivity) approximations. Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively $k$-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.

Topological Data Analysis (TDA) offers a suite of computational tools that provide quantified shape features in high dimensional data that can be used by modern statistical and predictive machine learning (ML) models. In particular, persistent homology (PH) takes in data (e.g., point clouds, images, time series) and derives compact representations of latent topological structures, known as persistence diagrams (PDs). Because PDs enjoy inherent noise tolerance, are interpretable and provide a solid basis for data analysis, and can be made compatible with the expansive set of well-established ML model architectures, PH has been widely adopted for model development including on sensitive data, such as genomic, cancer, sensor network, and financial data. Thus, TDA should be incorporated into secure end-to-end data analysis pipelines. In this paper, we take the first step to address this challenge and develop a version of the fundamental algorithm to compute PH on encrypted data using homomorphic encryption (HE).

Mutual coherence is a measure of similarity between two opinions. Although the notion comes from philosophy, it is essential for a wide range of technologies, e.g., the Wahl-O-Mat system. In Germany, this system helps voters to find candidates that are the closest to their political preferences. The exact computation of mutual coherence is highly time-consuming due to the iteration over all subsets of an opinion. Moreover, for every subset, an instance of the SAT model counting problem has to be solved which is known to be a hard problem in computer science. This work is the first study to accelerate this computation. We model the distribution of the so-called confirmation values as a mixture of three Gaussians and present efficient heuristics to estimate its model parameters. The mutual coherence is then approximated with the expected value of the distribution. Some of the presented algorithms are fully polynomial-time, others only require solving a small number of instances of the SAT model counting problem. The average squared error of our best algorithm lies below 0.0035 which is insignificant if the efficiency is taken into account. Furthermore, the accuracy is precise enough to be used in Wahl-O-Mat-like systems.