The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution

Subspace-based signal processing techniques, such as the Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT) algorithm, are popular methods for spectral estimation. These algorithms can achieve the so-called super-resolution scaling under low noise conditions, surpassing the well-known Nyquist limit. However, the performance of these algorithms under high-noise conditions is not as well understood. Existing state-of-the-art analysis indicates that ESPRIT and related algorithms can be resilient even for signals where each observation is corrupted by statistically independent, mean-zero noise of size $\mathcal{O}(1)$, but these analyses only show that the error $\epsilon$ decays at a slow rate $\epsilon=\mathcal{\tilde{O}}(n^{-1/2})$ with respect to the cutoff frequency $n$. In this work, we prove that under certain assumptions of bias and high noise, the ESPRIT algorithm can attain a significantly improved error scaling $\epsilon = \mathcal{\tilde{O}}(n^{-3/2})$, exhibiting noisy super-resolution scaling beyond the Nyquist limit. We further establish a theoretical lower bound and show that this scaling is optimal. Our analysis introduces novel matrix perturbation results, which could be of independent interest.

Energy Allocation for Multi-User Cooperative Molecular Communication Systems in the Internet of Bio-Nano Things

Cooperative molecular communication (MC) is a promising technology for facilitating communication between nanomachines in the Internet of Bio-Nano Things (IoBNT) field. However, the performance of IoBNT is limited by the availability of energy for cooperative MC. This paper presents a novel transmitter design scheme that utilizes molecule movement between reservoirs, creating concentration differences through the consumption of free energy, and encoding information on molecule types. The performance of the transmitter is primarily influenced by energy costs, which directly impact the overall IoBNT system performance. To address this, the paper focuses on optimizing energy allocation in cooperative MC for enhanced transmitter performance. Theoretical analysis is conducted for two transmitters. For scenarios with more than two users, a genetic algorithm is employed in the energy allocation to minimize the total bit error rate (BER). Finally, numerical results show the effectiveness of the proposed energy allocation strategies in the considered cooperative MC system.

CSST Strong Lensing Preparation: a Framework for Detecting Strong Lenses in the Multi-color Imaging Survey by the China Survey Space Telescope (CSST)

Strong gravitational lensing is a powerful tool for investigating dark matter and dark energy properties. With the advent of large-scale sky surveys, we can discover strong lensing systems on an unprecedented scale, which requires efficient tools to extract them from billions of astronomical objects. The existing mainstream lens-finding tools are based on machine learning algorithms and applied to cut-out-centered galaxies. However, according to the design and survey strategy of optical surveys by CSST, preparing cutouts with multiple bands requires considerable efforts. To overcome these challenges, we have developed a framework based on a hierarchical visual Transformer with a sliding window technique to search for strong lensing systems within entire images. Moreover, given that multi-color images of strong lensing systems can provide insights into their physical characteristics, our framework is specifically crafted to identify strong lensing systems in images with any number of channels. As evaluated using CSST mock data based on an Semi-Analytic Model named CosmoDC2, our framework achieves precision and recall rates of 0.98 and 0.90, respectively. To evaluate the effectiveness of our method in real observations, we have applied it to a subset of images from the DESI Legacy Imaging Surveys and media images from Euclid Early Release Observations. 61 new strong lensing system candidates are discovered by our method. However, we also identified false positives arising primarily from the simplified galaxy morphology assumptions within the simulation. This underscores the practical limitations of our approach while simultaneously highlighting potential avenues for future improvements.

A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions

Neural network wavefunctions optimized using the variational Monte Carlo method have been shown to produce highly accurate results for the electronic structure of atoms and small molecules, but the high cost of optimizing such wavefunctions prevents their application to larger systems. We propose the Subsampled Projected-Increment Natural Gradient Descent (SPRING) optimizer to reduce this bottleneck. SPRING combines ideas from the recently introduced minimum-step stochastic reconfiguration optimizer (MinSR) and the classical randomized Kaczmarz method for solving linear least-squares problems. We demonstrate that SPRING outperforms both MinSR and the popular Kronecker-Factored Approximate Curvature method (KFAC) across a number of small atoms and molecules, given that the learning rates of all methods are optimally tuned. For example, on the oxygen atom, SPRING attains chemical accuracy after forty thousand training iterations, whereas both MinSR and KFAC fail to do so even after one hundred thousand iterations.

Convergence of stochastic gradient descent on parameterized sphere with applications to variational Monte Carlo simulation

We analyze stochastic gradient descent (SGD) type algorithms on a high-dimensional sphere which is parameterized by a neural network up to a normalization constant. We provide a new algorithm for the setting of supervised learning and show its convergence both theoretically and numerically. We also provide the first proof of convergence for the unsupervised setting, which corresponds to the widely used variational Monte Carlo (VMC) method in quantum physics.

Anti-symmetric Barron functions and their approximation with sums of determinants

A fundamental problem in quantum physics is to encode functions that are completely anti-symmetric under permutations of identical particles. The Barron space consists of high-dimensional functions that can be parameterized by infinite neural networks with one hidden layer. By explicitly encoding the anti-symmetric structure, we prove that the anti-symmetric functions which belong to the Barron space can be efficiently approximated with sums of determinants. This yields a factorial improvement in complexity compared to the standard representation in the Barron space and provides a theoretical explanation for the effectiveness of determinant-based architectures in ab-initio quantum chemistry.

Personalized Pricing with Invalid Instrumental Variables: Identification, Estimation, and Policy Learning

Pricing based on individual customer characteristics is widely used to maximize sellers' revenues. This work studies offline personalized pricing under endogeneity using an instrumental variable approach. Standard instrumental variable methods in causal inference/econometrics either focus on a discrete treatment space or require the exclusion restriction of instruments from having a direct effect on the outcome, which limits their applicability in personalized pricing. In this paper, we propose a new policy learning method for Personalized pRicing using Invalid iNsTrumental variables (PRINT) for continuous treatment that allow direct effects on the outcome. Specifically, relying on the structural models of revenue and price, we establish the identifiability condition of an optimal pricing strategy under endogeneity with the help of invalid instrumental variables. Based on this new identification, which leads to solving conditional moment restrictions with generalized residual functions, we construct an adversarial min-max estimator and learn an optimal pricing strategy. Furthermore, we establish an asymptotic regret bound to find an optimal pricing strategy. Finally, we demonstrate the effectiveness of the proposed method via extensive simulation studies as well as a real data application from an US online auto loan company.

Probabilistic Model Incorporating Auxiliary Covariates to Control FDR

Controlling False Discovery Rate (FDR) while leveraging the side information of multiple hypothesis testing is an emerging research topic in modern data science. Existing methods rely on the test-level covariates while ignoring metrics about test-level covariates. This strategy may not be optimal for complex large-scale problems, where indirect relations often exist among test-level covariates and auxiliary metrics or covariates. We incorporate auxiliary covariates among test-level covariates in a deep Black-Box framework controlling FDR (named as NeurT-FDR) which boosts statistical power and controls FDR for multiple-hypothesis testing. Our method parametrizes the test-level covariates as a neural network and adjusts the auxiliary covariates through a regression framework, which enables flexible handling of high-dimensional features as well as efficient end-to-end optimization. We show that NeurT-FDR makes substantially more discoveries in three real datasets compared to competitive baselines.

Taming the sign problem of explicitly antisymmetrized neural networks via rough activation functions

Explicit antisymmetrization of a two-layer neural network is a potential candidate for a universal function approximator for generic antisymmetric functions, which are ubiquitous in quantum physics. However, this strategy suffers from a sign problem, namely, due to near exact cancellation of positive and negative contributions, the magnitude of the antisymmetrized function may be significantly smaller than that before antisymmetrization. We prove that the severity of the sign problem is directly related to the smoothness of the activation function. For smooth activation functions (e.g., $\tanh$), the sign problem of the explicitly antisymmetrized two-layer neural network deteriorates super-polynomially with respect to the system size. On the other hand, for rough activation functions (e.g., ReLU), the deterioration rate of the sign problem can be tamed to be at most polynomial with respect to the system size. Finally, the cost of a direct implementation of antisymmetrized two-layer neural network scales factorially with respect to the system size. We describe an efficient algorithm for approximate evaluation of such a network, of which the cost scales polynomially with respect to the system size and inverse precision.

Monte Carlo Tree Search based Hybrid Optimization of Variational Quantum Circuits

Variational quantum algorithms stand at the forefront of simulations on near-term and future fault-tolerant quantum devices. While most variational quantum algorithms involve only continuous optimization variables, the representational power of the variational ansatz can sometimes be significantly enhanced by adding certain discrete optimization variables, as is exemplified by the generalized quantum approximate optimization algorithm (QAOA). However, the hybrid discrete-continuous optimization problem in the generalized QAOA poses a challenge to the optimization. We propose a new algorithm called MCTS-QAOA, which combines a Monte Carlo tree search method with an improved natural policy gradient solver to optimize the discrete and continuous variables in the quantum circuit, respectively. We find that MCTS-QAOA has excellent noise-resilience properties and outperforms prior algorithms in challenging instances of the generalized QAOA.