Cluster analysis requires many decisions: the clustering method and the implied reference model, the number of clusters and, often, several hyper-parameters and algorithms' tunings. In practice, one produces several partitions, and a final one is chosen based on validation or selection criteria. There exist an abundance of validation methods that, implicitly or explicitly, assume a certain clustering notion. Moreover, they are often restricted to operate on partitions obtained from a specific method. In this paper, we focus on groups that can be well separated by quadratic or linear boundaries. The reference cluster concept is defined through the quadratic discriminant score function and parameters describing clusters' size, center and scatter. We develop two cluster-quality criteria called quadratic scores. We show that these criteria are consistent with groups generated from a general class of elliptically-symmetric distributions. The quest for this type of groups is common in applications. The connection with likelihood theory for mixture models and model-based clustering is investigated. Based on bootstrap resampling of the quadratic scores, we propose a selection rule that allows choosing among many clustering solutions. The proposed method has the distinctive advantage that it can compare partitions that cannot be compared with other state-of-the-art methods. Extensive numerical experiments and the analysis of real data show that, even if some competing methods turn out to be superior in some setups, the proposed methodology achieves a better overall performance.
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Nowadays, numerical models are widely used in most of engineering fields to simulate the behaviour of complex systems, such as for example power plants or wind turbine in the energy sector. Those models are nevertheless affected by uncertainty of different nature (numerical, epistemic) which can affect the reliability of their predictions. We develop here a new method for quantifying conditional parameter uncertainty within a chain of two numerical models in the context of multiphysics simulation. More precisely, we aim to calibrate the parameters $\theta$ of the second model of the chain conditionally on the value of parameters $\lambda$ of the first model, while assuming the probability distribution of $\lambda$ is known. This conditional calibration is carried out from the available experimental data of the second model. In doing so, we aim to quantify as well as possible the impact of the uncertainty of $\lambda$ on the uncertainty of $\theta$. To perform this conditional calibration, we set out a nonparametric Bayesian formalism to estimate the functional dependence between $\theta$ and $\lambda$, denoted by $\theta(\lambda)$. First, each component of $\theta(\lambda)$ is assumed to be the realization of a Gaussian process prior. Then, if the second model is written as a linear function of $\theta(\lambda)$, the Bayesian machinery allows us to compute analytically the posterior predictive distribution of $\theta(\lambda)$ for any set of realizations $\lambda$. The effectiveness of the proposed method is illustrated on several analytical examples.
Preference-based optimization algorithms are iterative procedures that seek the optimal calibration of a decision vector based only on comparisons between couples of different tunings. At each iteration, a human decision-maker expresses a preference between two calibrations (samples), highlighting which one, if any, is better than the other. The optimization procedure must use the observed preferences to find the tuning of the decision vector that is most preferred by the decision-maker, while also minimizing the number of comparisons. In this work, we formulate the preference-based optimization problem from a utility theory perspective. Then, we propose GLISp-r, an extension of a recent preference-based optimization procedure called GLISp. The latter uses a Radial Basis Function surrogate to describe the tastes of the decision-maker. Iteratively, GLISp proposes new samples to compare with the best calibration available by trading off exploitation of the surrogate model and exploration of the decision space. In GLISp-r, we propose a different criterion to use when looking for new candidate samples that is inspired by MSRS, a popular procedure in the black-box optimization framework. Compared to GLISp, GLISp-r is less likely to get stuck on local optima of the preference-based optimization problem. We motivate this claim theoretically, with a proof of global convergence, and empirically, by comparing the performances of GLISp and GLISp-r on several benchmark optimization problems.
The effectiveness of graphical recommender system depends on the quantity and quality of negative sampling. This paper selects some typical recommender system models, as well as some latest negative sampling strategies on the models as baseline. Based on typical graphical recommender model, we divide sample region into assigned-n areas and use AdaSim to give different weight to these areas to form positive set and negative set. Because of the volume and significance of negative items, we also proposed a subset selection model to narrow the core negative samples.
Branching process inspired models are widely used to estimate the effective reproduction number -- a useful summary statistic describing an infectious disease outbreak -- using counts of new cases. Case data is a real-time indicator of changes in the reproduction number, but is challenging to work with because cases fluctuate due to factors unrelated to the number of new infections. We develop a new model that incorporates the number of diagnostic tests as a surveillance model covariate. Using simulated data and data from the SARS-CoV-2 pandemic in California, we demonstrate that incorporating tests leads to improved performance over the state-of-the-art.
Data-driven algorithm design is a promising, learning-based approach for beyond worst-case analysis of algorithms with tunable parameters. An important open problem is the design of computationally efficient data-driven algorithms for combinatorial algorithm families with multiple parameters. As one fixes the problem instance and varies the parameters, the "dual" loss function typically has a piecewise-decomposable structure, i.e. is well-behaved except at certain sharp transition boundaries. In this work we initiate the study of techniques to develop efficient ERM learning algorithms for data-driven algorithm design by enumerating the pieces of the sum dual loss functions for a collection of problem instances. The running time of our approach scales with the actual number of pieces that appear as opposed to worst case upper bounds on the number of pieces. Our approach involves two novel ingredients -- an output-sensitive algorithm for enumerating polytopes induced by a set of hyperplanes using tools from computational geometry, and an execution graph which compactly represents all the states the algorithm could attain for all possible parameter values. We illustrate our techniques by giving algorithms for pricing problems, linkage-based clustering and dynamic-programming based sequence alignment.
Transient errors from the dynamic NISQ noise landscape are challenging to comprehend and are especially detrimental to classes of applications that are iterative and/or long-running, and therefore their timely mitigation is important for quantum advantage in real-world applications. The most popular examples of iterative long-running quantum applications are variational quantum algorithms (VQAs). Iteratively, VQA's classical optimizer evaluates circuit candidates on an objective function and picks the best circuits towards achieving the application's target. Noise fluctuation can cause a significant transient impact on the objective function estimation of the VQA iterations / tuning candidates. This can severely affect VQA tuning and, by extension, its accuracy and convergence. This paper proposes QISMET: Quantum Iteration Skipping to Mitigate Error Transients, to navigate the dynamic noise landscape of VQAs. QISMET actively avoids instances of high fluctuating noise which are predicted to have a significant transient error impact on specific VQA iterations. To achieve this, QISMET estimates transient error in VQA iterations and designs a controller to keep the VQA tuning faithful to the transient-free scenario. By doing so, QISMET efficiently mitigates a large portion of the transient noise impact on VQAs and is able to improve the fidelity by 1.3x-3x over a traditional VQA baseline, with 1.6-2.4x improvement over alternative approaches, across different applications and machines. Further, to diligently analyze the effects of transients, this work also builds transient noise models for target VQA applications from observing real machine transients. These are then integrated with the Qiskit simulator.
Image recognition is one of the primary applications of machine learning algorithms. Nevertheless, machine learning models used in modern image recognition systems consist of millions of parameters that usually require significant computational time to be adjusted. Moreover, adjustment of model hyperparameters leads to additional overhead. Because of this, new developments in machine learning models and hyperparameter optimization techniques are required. This paper presents a quantum-inspired hyperparameter optimization technique and a hybrid quantum-classical machine learning model for supervised learning. We benchmark our hyperparameter optimization method over standard black-box objective functions and observe performance improvements in the form of reduced expected run times and fitness in response to the growth in the size of the search space. We test our approaches in a car image classification task and demonstrate a full-scale implementation of the hybrid quantum ResNet model with the tensor train hyperparameter optimization. Our tests show a qualitative and quantitative advantage over the corresponding standard classical tabular grid search approach used with a deep neural network ResNet34. A classification accuracy of 0.97 was obtained by the hybrid model after 18 iterations, whereas the classical model achieved an accuracy of 0.92 after 75 iterations.
Bayesian optimization (BO), while proved highly effective for many black-box function optimization tasks, requires practitioners to carefully select priors that well model their functions of interest. Rather than specifying by hand, researchers have investigated transfer learning based methods to automatically learn the priors, e.g. multi-task BO (Swersky et al., 2013), few-shot BO (Wistuba and Grabocka, 2021) and HyperBO (Wang et al., 2022). However, those prior learning methods typically assume that the input domains are the same for all tasks, weakening their ability to use observations on functions with different domains or generalize the learned priors to BO on different search spaces. In this work, we present HyperBO+: a pre-training approach for hierarchical Gaussian processes that enables the same prior to work universally for Bayesian optimization on functions with different domains. We propose a two-step pre-training method and analyze its appealing asymptotic properties and benefits to BO both theoretically and empirically. On real-world hyperparameter tuning tasks that involve multiple search spaces, we demonstrate that HyperBO+ is able to generalize to unseen search spaces and achieves lower regrets than competitive baselines.
We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more precise result, we show that if any hereditary graph class $\mathcal{G}$ admits $O(n^{1-\epsilon})$ separators, then for any fixed $\delta\in(0,\epsilon)$ every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-depth and a complete graph of size $O(n^{1-\epsilon+\delta})$. This result holds with $\delta=0$ if we allow $H$ to have tree-depth $O(\log\log n)$. Moreover, using extensions of classical isoperimetric inequalties for grids graphs, we show the dependence on $\delta$ in our results and the above $\text{td}(H)\in O(\log\log n)$ bound are both best possible. We prove that $n$-vertex graphs of bounded treewidth are subgraphs of the product of a graph with tree-depth $t$ and a complete graph of size $O(n^{1/t})$, which is best possible. Finally, we investigate the conjecture that for any hereditary graph class $\mathcal{G}$ that admits $O(n^{1-\epsilon})$ separators, every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-width and a complete graph of size $O(n^{1-\epsilon})$. We prove this for various classes $\mathcal{G}$ of interest.
We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.
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