In this work, we solve differential equations using quantum Chebyshev feature maps. We propose a tensor product over a summation of Pauli-Z operators as a change in the measurement observables resulting in improved accuracy and reduced computation time for initial value problems processed by floating boundary handling. This idea has been tested on solving the complex dynamics of a Riccati equation as well as on a system of differential equations. Furthermore, a second-order differential equation is investigated in which we propose adding entangling layers to improve accuracy without increasing the variational parameters. Additionally, a modified self-adaptivity approach of physics-informed neural networks is incorporated to balance the multi-objective loss function. Finally, a new quantum circuit structure is proposed to approximate multivariable functions, tested on solving a 2D Poisson's equation.
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We construct a randomized vector quantizer which has a smaller maximum error compared to all known lattice quantizers with the same entropy for dimensions 5, 6, ..., 48, and also has a smaller mean squared error compared to known lattice quantizers with the same entropy for dimensions 35, ..., 48, in the high resolution limit. Moreover, our randomized quantizer has a desirable property that the quantization error is always uniform over the ball and independent of the input. Our construction is based on applying rejection sampling on universal quantization, which allows us to shape the error distribution to be any continuous distribution, not only uniform distributions over basic cells of a lattice as in conventional dithered quantization. We also characterize the high SNR limit of one-shot channel simulation for any additive noise channel under a mild assumption (e.g., the AWGN channel), up to an additive constant of 1.45 bits.
Modeling complementary relationships greatly helps recommender systems to accurately and promptly recommend the subsequent items when one item is purchased. Unlike traditional similar relationships, items with complementary relationships may be purchased successively (such as iPhone and Airpods Pro), and they not only share relevance but also exhibit dissimilarity. Since the two attributes are opposites, modeling complementary relationships is challenging. Previous attempts to exploit these relationships have either ignored or oversimplified the dissimilarity attribute, resulting in ineffective modeling and an inability to balance the two attributes. Since Graph Neural Networks (GNNs) can capture the relevance and dissimilarity between nodes in the spectral domain, we can leverage spectral-based GNNs to effectively understand and model complementary relationships. In this study, we present a novel approach called Spectral-based Complementary Graph Neural Networks (SComGNN) that utilizes the spectral properties of complementary item graphs. We make the first observation that complementary relationships consist of low-frequency and mid-frequency components, corresponding to the relevance and dissimilarity attributes, respectively. Based on this spectral observation, we design spectral graph convolutional networks with low-pass and mid-pass filters to capture the low-frequency and mid-frequency components. Additionally, we propose a two-stage attention mechanism to adaptively integrate and balance the two attributes. Experimental results on four e-commerce datasets demonstrate the effectiveness of our model, with SComGNN significantly outperforming existing baseline models.
Denoising diffusion probabilistic models (DDPMs) have recently taken the field of generative modeling by storm, pioneering new state-of-the-art results in disciplines such as computer vision and computational biology for diverse tasks ranging from text-guided image generation to structure-guided protein design. Along this latter line of research, methods have recently been proposed for generating 3D molecules using equivariant graph neural networks (GNNs) within a DDPM framework. However, such methods are unable to learn important geometric and physical properties of 3D molecules during molecular graph generation, as they adopt molecule-agnostic and non-geometric GNNs as their 3D graph denoising networks, which negatively impacts their ability to effectively scale to datasets of large 3D molecules. In this work, we address these gaps by introducing the Geometry-Complete Diffusion Model (GCDM) for 3D molecule generation, which outperforms existing 3D molecular diffusion models by significant margins across conditional and unconditional settings for the QM9 dataset as well as for the larger GEOM-Drugs dataset. Importantly, we demonstrate that the geometry-complete denoising process GCDM learns for 3D molecule generation allows the model to generate realistic and stable large molecules at the scale of GEOM-Drugs, whereas previous methods fail to do so with the features they learn. Additionally, we show that extensions of GCDM can not only effectively design 3D molecules for specific protein pockets but also that GCDM's geometric features can effectively be repurposed to directly optimize the geometry and chemical composition of existing 3D molecules for specific molecular properties, demonstrating new, real-world versatility of molecular diffusion models. Our source code and data are freely available at //github.com/BioinfoMachineLearning/Bio-Diffusion.
Ordinary differential equation (ODE) is an important tool to study the dynamics of a system of biological and physical processes. A central question in ODE modeling is to infer the significance of individual regulatory effect of one signal variable on another. However, building confidence band for ODE with unknown regulatory relations is challenging, and it remains largely an open question. In this article, we construct post-regularization confidence band for individual regulatory function in ODE with unknown functionals and noisy data observations. Our proposal is the first of its kind, and is built on two novel ingredients. The first is a new localized kernel learning approach that combines reproducing kernel learning with local Taylor approximation, and the second is a new de-biasing method that tackles infinite-dimensional functionals and additional measurement errors. We show that the constructed confidence band has the desired asymptotic coverage probability, and the recovered regulatory network approaches the truth with probability tending to one. We establish the theoretical properties when the number of variables in the system can be either smaller or larger than the number of sampling time points, and we study the regime-switching phenomenon. We demonstrate the efficacy of the proposed method through both simulations and illustrations with two data applications.
Efficient implementation of massive multiple-input-multiple-output (MIMO) transceivers is essential for the next-generation wireless networks. To reduce the high computational complexity of the massive MIMO transceiver, in this paper, we propose a new massive MIMO architecture using finite-precision arithmetic. First, we conduct the rounding error analysis and derive the lower bound of the achievable rate for single-input-multiple-output (SIMO) using maximal ratio combining (MRC) and multiple-input-single-output (MISO) systems using maximal ratio transmission (MRT) with finite-precision arithmetic. Then, considering the multi-user scenario, the rounding error analysis of zero-forcing (ZF) detection and precoding is derived by using the normal equations (NE) method. The corresponding lower bounds of the achievable sum rate are also derived and asymptotic analyses are presented. Built upon insights from these analyses and lower bounds, we propose a mixed-precision architecture for massive MIMO systems to offset performance gaps due to finite-precision arithmetic. The corresponding analysis of rounding errors and computational costs is obtained. Simulation results validate the derived bounds and underscore the superiority of the proposed mixed-precision architecture to the conventional structure.
We develop a mesh-based semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions recast as a non-linear transport problem for a momentum 1-form. A linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing viscosity limit and is applicable to inviscid incompressible Euler flows. Conservation of energy and helicity are enforced separately.
Understanding the importance of the inputs on the output is useful across many tasks. This work provides an information-theoretic framework to analyse the influence of inputs for text classification tasks. Natural language processing (NLP) tasks take either a single element input or multiple element inputs to predict an output variable, where an element is a block of text. Each text element has two components: an associated semantic meaning and a linguistic realization. Multiple-choice reading comprehension (MCRC) and sentiment classification (SC) are selected to showcase the framework. For MCRC, it is found that the context influence on the output compared to the question influence reduces on more challenging datasets. In particular, more challenging contexts allow a greater variation in complexity of questions. Hence, test creators need to carefully consider the choice of the context when designing multiple-choice questions for assessment. For SC, it is found the semantic meaning of the input text dominates (above 80\% for all datasets considered) compared to its linguistic realisation when determining the sentiment. The framework is made available at: //github.com/WangLuran/nlp-element-influence
This paper presents a groundbreaking approach to causal inference by integrating continuous normalizing flows (CNFs) with parametric submodels, enhancing their geometric sensitivity and improving upon traditional Targeted Maximum Likelihood Estimation (TMLE). Our method employs CNFs to refine TMLE, optimizing the Cram\'er-Rao bound and transitioning from a predefined distribution $p_0$ to a data-driven distribution $p_1$. We innovate further by embedding Wasserstein gradient flows within Fokker-Planck equations, thus imposing geometric structures that boost the robustness of CNFs, particularly in optimal transport theory. Our approach addresses the disparity between sample and population distributions, a critical factor in parameter estimation bias. We leverage optimal transport and Wasserstein gradient flows to develop causal inference methodologies with minimal variance in finite-sample settings, outperforming traditional methods like TMLE and AIPW. This novel framework, centered on Wasserstein gradient flows, minimizes variance in efficient influence functions under distribution $p_t$. Preliminary experiments showcase our method's superiority, yielding lower mean-squared errors compared to standard flows, thereby demonstrating the potential of geometry-aware normalizing Wasserstein flows in advancing statistical modeling and inference.
Predictive multiplicity refers to the phenomenon in which classification tasks may admit multiple competing models that achieve almost-equally-optimal performance, yet generate conflicting outputs for individual samples. This presents significant concerns, as it can potentially result in systemic exclusion, inexplicable discrimination, and unfairness in practical applications. Measuring and mitigating predictive multiplicity, however, is computationally challenging due to the need to explore all such almost-equally-optimal models, known as the Rashomon set, in potentially huge hypothesis spaces. To address this challenge, we propose a novel framework that utilizes dropout techniques for exploring models in the Rashomon set. We provide rigorous theoretical derivations to connect the dropout parameters to properties of the Rashomon set, and empirically evaluate our framework through extensive experimentation. Numerical results show that our technique consistently outperforms baselines in terms of the effectiveness of predictive multiplicity metric estimation, with runtime speedup up to $20\times \sim 5000\times$. With efficient Rashomon set exploration and metric estimation, mitigation of predictive multiplicity is then achieved through dropout ensemble and model selection.
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.
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