The development of variational quantum algorithms is crucial for the application of NISQ computers. Such algorithms require short quantum circuits, which are more amenable to implementation on near-term hardware, and many such methods have been developed. One of particular interest is the so-called the variational diagonalization method, which constitutes an important algorithmic subroutine, and it can be used directly for working with data encoded in quantum states. In particular, it can be applied to discern the features of quantum states, such as entanglement properties of a system, or in quantum machine learning algorithms. In this work, we tackle the problem of designing a very shallow quantum circuit, required in the quantum state diagonalization task, by utilizing reinforcement learning. To achieve this, we utilize a novel encoding method that can be used to tackle the problem of circuit depth optimization using a reinforcement learning approach. We demonstrate that our approach provides a solid approximation to the diagonalization task while using a small number of gates. The circuits proposed by the reinforcement learning methods are shallower than the standard variational quantum state diagonalization algorithm, and thus can be used in situations where the depth of quantum circuits is limited by the hardware capabilities.
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We address the computational efficiency in solving the A-optimal Bayesian design of experiments problems for which the observational map is based on partial differential equations and, consequently, is computationally expensive to evaluate. A-optimality is a widely used and easy-to-interpret criterion for Bayesian experimental design. This criterion seeks the optimal experimental design by minimizing the expected conditional variance, which is also known as the expected posterior variance. This study presents a novel likelihood-free approach to the A-optimal experimental design that does not require sampling or integrating the Bayesian posterior distribution. The expected conditional variance is obtained via the variance of the conditional expectation using the law of total variance, and we take advantage of the orthogonal projection property to approximate the conditional expectation. We derive an asymptotic error estimation for the proposed estimator of the expected conditional variance and show that the intractability of the posterior distribution does not affect the performance of our approach. We use an artificial neural network (ANN) to approximate the nonlinear conditional expectation in the implementation of our method. We then extend our approach for dealing with the case that the domain of experimental design parameters is continuous by integrating the training process of the ANN into minimizing the expected conditional variance. Through numerical experiments, we demonstrate that our method greatly reduces the number of observation model evaluations compared with widely used importance sampling-based approaches. This reduction is crucial, considering the high computational cost of the observational models. Code is available at //github.com/vinh-tr-hoang/DOEviaPACE.
Operator splitting is a popular divide-and-conquer strategy for solving differential equations. Typically, the right-hand side of the differential equation is split into a number of parts that are then integrated separately. Many methods are known that split the right-hand side into two parts. This approach is limiting, however, and there are situations when 3-splitting is more natural and ultimately more advantageous. The second-order Strang operator-splitting method readily generalizes to a right-hand side splitting into any number of operators. It is arguably the most popular method for 3-splitting because of its efficiency, ease of implementation, and intuitive nature. Other 3-splitting methods exist, but they are less well-known, and \rev{analysis and} evaluation of their performance in practice are scarce. We demonstrate the effectiveness of some alternative 3-split, second-order methods to Strang splitting on two problems: the reaction-diffusion Brusselator, which can be split into three parts that each have closed-form solutions, and the kinetic Vlasov--Poisson equations that is used in semi-Lagrangian plasma simulations. We find alternative second-order 3-operator-splitting methods that realize efficiency gains of 10\%--20\% over traditional Strang splitting. Our analysis for the practical assessment of efficiency of operator-splitting methods includes the computational cost of the integrators and can be used in method design.
Selective inference methods are developed for group lasso estimators for use with a wide class of distributions and loss functions. The method includes the use of exponential family distributions, as well as quasi-likelihood modeling for overdispersed count data, for example, and allows for categorical or grouped covariates as well as continuous covariates. A randomized group-regularized optimization problem is studied. The added randomization allows us to construct a post-selection likelihood which we show to be adequate for selective inference when conditioning on the event of the selection of the grouped covariates. This likelihood also provides a selective point estimator, accounting for the selection by the group lasso. Confidence regions for the regression parameters in the selected model take the form of Wald-type regions and are shown to have bounded volume. The selective inference method for grouped lasso is illustrated on data from the national health and nutrition examination survey while simulations showcase its behaviour and favorable comparison with other methods.
Stochastic inverse problems are typically encountered when it is wanted to quantify the uncertainty affecting the inputs of computer models. They consist in estimating input distributions from noisy, observable outputs, and such problems are increasingly examined in Bayesian contexts where the targeted inputs are affected by stochastic uncertainties. In this regard, a stochastic input can be qualified as meaningful if it explains most of the output uncertainty. While such inverse problems are characterized by identifiability conditions, constraints of "signal to noise", that can formalize this meaningfulness, should be accounted for within the definition of the model, prior to inference. This article investigates the possibility of forcing a solution to be meaningful in the context of parametric uncertainty quantification, through the tools of global sensitivity analysis and information theory (variance, entropy, Fisher information). Such forcings have mainly the nature of constraints placed on the input covariance, and can be made explicit by considering linear or linearizable models. Simulated experiments indicate that, when injected into the modeling process, these constraints can limit the influence of measurement or process noise on the estimation of the input distribution, and let hope for future extensions in a full non-linear framework, for example through the use of linear Gaussian mixtures.
Surgical instrument segmentation is recognised as a key enabler to provide advanced surgical assistance and improve computer assisted interventions. In this work, we propose SegMatch, a semi supervised learning method to reduce the need for expensive annotation for laparoscopic and robotic surgical images. SegMatch builds on FixMatch, a widespread semi supervised classification pipeline combining consistency regularization and pseudo labelling, and adapts it for the purpose of segmentation. In our proposed SegMatch, the unlabelled images are weakly augmented and fed into the segmentation model to generate a pseudo-label to enforce the unsupervised loss against the output of the model for the adversarial augmented image on the pixels with a high confidence score. Our adaptation for segmentation tasks includes carefully considering the equivariance and invariance properties of the augmentation functions we rely on. To increase the relevance of our augmentations, we depart from using only handcrafted augmentations and introduce a trainable adversarial augmentation strategy. Our algorithm was evaluated on the MICCAI Instrument Segmentation Challenge datasets Robust-MIS 2019 and EndoVis 2017. Our results demonstrate that adding unlabelled data for training purposes allows us to surpass the performance of fully supervised approaches which are limited by the availability of training data in these challenges. SegMatch also outperforms a range of state-of-the-art semi-supervised learning semantic segmentation models in different labelled to unlabelled data ratios.
This work puts forth low-complexity Riemannian subspace descent algorithms for the minimization of functions over the symmetric positive definite (SPD) manifold. Different from the existing Riemannian gradient descent variants, the proposed approach utilizes carefully chosen subspaces that allow the update to be written as a product of the Cholesky factor of the iterate and a sparse matrix. The resulting updates avoid the costly matrix operations like matrix exponentiation and dense matrix multiplication, which are generally required in almost all other Riemannian optimization algorithms on SPD manifold. We further identify a broad class of functions, arising in diverse applications, such as kernel matrix learning, covariance estimation of Gaussian distributions, maximum likelihood parameter estimation of elliptically contoured distributions, and parameter estimation in Gaussian mixture model problems, over which the Riemannian gradients can be calculated efficiently. The proposed uni-directional and multi-directional Riemannian subspace descent variants incur per-iteration complexities of $\mathcal{O}(n)$ and $\mathcal{O}(n^2)$ respectively, as compared to the $\mathcal{O}(n^3)$ or higher complexity incurred by all existing Riemannian gradient descent variants. The superior runtime and low per-iteration complexity of the proposed algorithms is also demonstrated via numerical tests on large-scale covariance estimation problems.
The aim of the current research is to analyse and discover, in a real context, behaviours, reactions and modes of interaction of social actors (people) with the humanoid robot Pepper. Indeed, we wanted to observe in a real, highly frequented context, the reactions and interactions of people with Pepper, placed in a shop window, through a systematic observation approach. The most interesting aspects of this research will be illustrated, bearing in mind that this is a preliminary analysis, therefore, not yet definitively concluded.
Very distinct strategies can be deployed to recognize and characterize an unknown environment or a shape. A recent and promising approach, especially in robotics, is to reduce the complexity of the exploratory units to a minimum. Here, we show that this frugal strategy can be taken to the extreme by exploiting the power of statistical geometry and introducing new invariant features. We show that an elementary robot devoid of any orientation or observation system, exploring randomly, can access global information about an environment such as the values of the explored area and perimeter. The explored shapes are of arbitrary geometry and may even non-connected. From a dictionary, this most simple robot can thus identify various shapes such as famous monuments and even read a text.
We consider an unknown multivariate function representing a system-such as a complex numerical simulator-taking both deterministic and uncertain inputs. Our objective is to estimate the set of deterministic inputs leading to outputs whose probability (with respect to the distribution of the uncertain inputs) of belonging to a given set is less than a given threshold. This problem, which we call Quantile Set Inversion (QSI), occurs for instance in the context of robust (reliability-based) optimization problems, when looking for the set of solutions that satisfy the constraints with sufficiently large probability. To solve the QSI problem, we propose a Bayesian strategy based on Gaussian process modeling and the Stepwise Uncertainty Reduction (SUR) principle, to sequentially choose the points at which the function should be evaluated to efficiently approximate the set of interest. We illustrate the performance and interest of the proposed SUR strategy through several numerical experiments.
The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.
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