Dictionary Learning-based Inpainting on Triangular Meshes

The problem of inpainting consists of filling missing or damaged regions in images and videos in such a way that the filling pattern does not produce artifacts that deviate from the original data. In addition to restoring the missing data, the inpainting technique can also be used to remove undesired objects. In this work, we address the problem of inpainting on surfaces through a new method based on dictionary learning and sparse coding. Our method learns the dictionary through the subdivision of the mesh into patches and rebuilds the mesh via a method of reconstruction inspired by the Non-local Means method on the computed sparse codes. One of the advantages of our method is that it is capable of filling the missing regions and simultaneously removes noise and enhances important features of the mesh. Moreover, the inpainting result is globally coherent as the representation based on the dictionaries captures all the geometric information in the transformed domain. We present two variations of the method: a direct one, in which the model is reconstructed and restored directly from the representation in the transformed domain and a second one, adaptive, in which the missing regions are recreated iteratively through the successive propagation of the sparse code computed in the hole boundaries, which guides the local reconstructions. The second method produces better results for large regions because the sparse codes of the patches are adapted according to the sparse codes of the boundary patches. Finally, we present and analyze experimental results that demonstrate the performance of our method compared to the literature

Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer

We compare the BFGS optimizer, ADAM and Natural Gradient Descent (NatGrad) in the context of Variational Quantum Eigensolvers (VQEs). We systematically analyze their performance on the QAOA ansatz for the Transverse Field Ising Model (TFIM) as well as on overparametrized circuits with the ability to break the symmetry of the Hamiltonian. The BFGS algorithm is frequently unable to find a global minimum for systems beyond about 20 spins and ADAM easily gets trapped in local minima. On the other hand, NatGrad shows stable performance on all considered system sizes, albeit at a significantly higher cost per epoch. In sharp contrast to most classical gradient based learning, the performance of all optimizers is found to decrease upon seemingly benign overparametrization of the ansatz class, with BFGS and ADAM failing more often and more severely than NatGrad. Additional tests for the Heisenberg XXZ model corroborate the accuracy problems of BFGS in high dimensions, but they reveal some shortcomings of NatGrad as well. Our results suggest that great care needs to be taken in the choice of gradient based optimizers and the parametrization for VQEs.Comment: 16 pages, 6 figures, 1 tabl

Neural networks and quantum many-body physics: exploring reciprocal benefits.

One of the main reasons why the physics of quantum many-body systems is hard lies in the curse of dimensionality: The number of states of such systems increases exponentially with the number of degrees of freedom involved. As a result, computations for realistic systems become intractable, and even numerical methods are limited to comparably small system sizes. Many efforts in modern physics research are therefore concerned with finding efficient representations of quantum states and clever approximations schemes that would allow them to characterize physical systems of interest. Meanwhile, Deep Learning (DL) has solved many non-scientific problems that have been unaccessible to conventional methods for a similar reason. The concept underlying DL is to extract knowledge from data by identifying patterns and regularities. The remarkable success of DL has excited many physicists about the prospect of leveraging its power to solve intractable problems in physics. At the same time, DL turned out to be an interesting complex many-body problem in itself. In contrast to its widespread empirical applications, the theoretical foundation of DL is strongly underdeveloped. In particular, as long as its decision-making process and result interpretability remain opaque, DL can not claim the status of a scientific tool. In this thesis, I explore the interface between DL and quantum many-body physics, and investigate DL both as a tool and as a subject of study. The first project presented here is a theory-based study of a fundamental open question about the role of width and the number of parameters in deep neural networks. In this work, we consider a DL setup for the image recognition task on standard benchmarking datasets. We combine controlled experiments with a theoretical analysis, including analytical calculations for a toy model. The other three works focus on the application of Restricted Boltzmann Machines as generative models for the task of wavefunction reconstruction from measurement data on a quantum many-body system. First, we implement this approach as a software package, making it available as a tool for experimentalists. Following the idea that physics problems can be used to characterize DL tools, we then use our extensive knowledge of this setup to conduct a systematic study of how the RBM complexity scales with the complexity of the physical system. Finally, in a follow-up study we focus on the effects of parameter pruning techniques on the RBM and its scaling behavior

Automatic Bayesian Density Analysis

Making sense of a dataset in an automatic and unsupervised fashion is a challenging problem in statistics and AI. Classical approaches for {exploratory data analysis} are usually not flexible enough to deal with the uncertainty inherent to real-world data: they are often restricted to fixed latent interaction models and homogeneous likelihoods; they are sensitive to missing, corrupt and anomalous data; moreover, their expressiveness generally comes at the price of intractable inference. As a result, supervision from statisticians is usually needed to find the right model for the data. However, since domain experts are not necessarily also experts in statistics, we propose Automatic Bayesian Density Analysis (ABDA) to make exploratory data analysis accessible at large. Specifically, ABDA allows for automatic and efficient missing value estimation, statistical data type and likelihood discovery, anomaly detection and dependency structure mining, on top of providing accurate density estimation. Extensive empirical evidence shows that ABDA is a suitable tool for automatic exploratory analysis of mixed continuous and discrete tabular data.Comment: In proceedings of the Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19