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Denoising is intuitively related to projection. Indeed, under the manifold hypothesis, adding random noise is approximately equivalent to orthogonal perturbation. Hence, learning to denoise is approximately learning to project. In this paper, we use this observation to reinterpret denoising diffusion models as approximate gradient descent applied to the Euclidean distance function. We then provide straight-forward convergence analysis of the DDIM sampler under simple assumptions on the projection-error of the denoiser. Finally, we propose a new sampler based on two simple modifications to DDIM using insights from our theoretical results. In as few as 5-10 function evaluations, our sampler achieves state-of-the-art FID scores on pretrained CIFAR-10 and CelebA models and can generate high quality samples on latent diffusion models.

The postulates of quantum mechanics impose only unitary transformations on quantum states, which is a severe limitation for quantum machine learning algorithms. Quantum Splines (QSplines) have recently been proposed to approximate quantum activation functions to introduce non-linearity in quantum algorithms. However, QSplines make use of the HHL as a subroutine and require a fault-tolerant quantum computer to be correctly implemented. This work proposes the Generalised Hybrid Quantum Splines (GHQSplines), a novel method for approximating non-linear quantum activation functions using hybrid quantum-classical computation. The GHQSplines overcome the highly demanding requirements of the original QSplines in terms of quantum hardware and can be implemented using near-term quantum computers. Furthermore, the proposed method relies on a flexible problem representation for non-linear approximation and it is suitable to be embedded in existing quantum neural network architectures. In addition, we provide a practical implementation of the GHQSplines using Pennylane and show that our model outperforms the original QSplines in terms of quality of fitting.

In the noisy intermediate-scale quantum era, variational quantum algorithms (VQAs) have emerged as a promising avenue to obtain quantum advantage. However, the success of VQAs depends on the expressive power of parameterised quantum circuits, which is constrained by the limited gate number and the presence of barren plateaus. In this work, we propose and numerically demonstrate a novel approach for VQAs, utilizing randomised quantum circuits to generate the variational wavefunction. We parameterize the distribution function of these random circuits using artificial neural networks and optimize it to find the solution. This random-circuit approach presents a trade-off between the expressive power of the variational wavefunction and time cost, in terms of the sampling cost of quantum circuits. Given a fixed gate number, we can systematically increase the expressive power by extending the quantum-computing time. With a sufficiently large permissible time cost, the variational wavefunction can approximate any quantum state with arbitrary accuracy. Furthermore, we establish explicit relationships between expressive power, time cost, and gate number for variational quantum eigensolvers. These results highlight the promising potential of the random-circuit approach in achieving a high expressive power in quantum computing.

Traditional partial differential equation (PDE) solvers can be computationally expensive, which motivates the development of faster methods, such as reduced-order-models (ROMs). We present GPLaSDI, a hybrid deep-learning and Bayesian ROM. GPLaSDI trains an autoencoder on full-order-model (FOM) data and simultaneously learns simpler equations governing the latent space. These equations are interpolated with Gaussian Processes, allowing for uncertainty quantification and active learning, even with limited access to the FOM solver. Our framework is able to achieve up to 100,000 times speed-up and less than 7% relative error on fluid mechanics problems.

Non-autoregressive models have been widely studied in the Complete Information Scenario (CIS), in which the input has complete information of corresponding output. However, their explorations in the Incomplete Information Scenario (IIS) are extremely limited. Our analyses reveal that the IIS's incomplete input information will augment the inherent limitations of existing non-autoregressive models trained under Maximum Likelihood Estimation. In this paper, we propose for the IIS an Adversarial Non-autoregressive Transformer (ANT) which has two features: 1) Position-Aware Self-Modulation to provide more reasonable hidden representations, and 2) Dependency Feed Forward Network to strengthen its capacity in dependency modeling. We compare ANT with other mainstream models in the IIS and demonstrate that ANT can achieve comparable performance with much fewer decoding iterations. Furthermore, we show its great potential in various applications like latent interpolation and semi-supervised learning.

Accurate hand gesture prediction is crucial for effective upper-limb prosthetic limbs control. As the high flexibility and multiple degrees of freedom exhibited by human hands, there has been a growing interest in integrating deep networks with high-density surface electromyography (HD-sEMG) grids to enhance gesture recognition capabilities. However, many existing methods fall short in fully exploit the specific spatial topology and temporal dependencies present in HD-sEMG data. Additionally, these studies are often limited number of gestures and lack generality. Hence, this study introduces a novel gesture recognition method, named STGCN-GR, which leverages spatio-temporal graph convolution networks for HD-sEMG-based human-machine interfaces. Firstly, we construct muscle networks based on functional connectivity between channels, creating a graph representation of HD-sEMG recordings. Subsequently, a temporal convolution module is applied to capture the temporal dependences in the HD-sEMG series and a spatial graph convolution module is employed to effectively learn the intrinsic spatial topology information among distinct HD-sEMG channels. We evaluate our proposed model on a public HD-sEMG dataset comprising a substantial number of gestures (i.e., 65). Our results demonstrate the remarkable capability of the STGCN-GR method, achieving an impressive accuracy of 91.07% in predicting gestures, which surpasses state-of-the-art deep learning methods applied to the same dataset.

We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.

We examine the linear regression problem in a challenging high-dimensional setting with correlated predictors to explain and predict relevant quantities, with explicitly allowing the regression coefficient to vary from sparse to dense. Most classical high-dimensional regression estimators require some degree of sparsity. We discuss the more recent concepts of variable screening and random projection as computationally fast dimension reduction tools, and propose a new random projection matrix tailored to the linear regression problem with a theoretical bound on the gain in expected prediction error over conventional random projections. Around this new random projection, we built the Sparse Projected Averaged Regression (SPAR) method combining probabilistic variable screening steps with the random projection steps to obtain an ensemble of small linear models. In difference to existing methods, we introduce a thresholding parameter to obtain some degree of sparsity. In extensive simulations and two real data applications we guide through the elements of this method and compare prediction and variable selection performance to various competitors. For prediction, our method performs at least as good as the best competitors in most settings with a high number of truly active variables, while variable selection remains a hard task for all methods in high dimensions.

The problem of Multiple Object Tracking (MOT) consists in following the trajectory of different objects in a sequence, usually a video. In recent years, with the rise of Deep Learning, the algorithms that provide a solution to this problem have benefited from the representational power of deep models. This paper provides a comprehensive survey on works that employ Deep Learning models to solve the task of MOT on single-camera videos. Four main steps in MOT algorithms are identified, and an in-depth review of how Deep Learning was employed in each one of these stages is presented. A complete experimental comparison of the presented works on the three MOTChallenge datasets is also provided, identifying a number of similarities among the top-performing methods and presenting some possible future research directions.

Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis, thereby allowing manual manipulation in predicting the final answer.