DeepTTV: Deep Learning Prediction of Hidden Exoplanet From Transit Timing Variations

Transit timing variation (TTV) provides rich information about the mass and orbital properties of exoplanets, which are often obtained by solving an inverse problem via Markov Chain Monte Carlo (MCMC). In this paper, we design a new data-driven approach, which potentially can be applied to problems that are hard to traditional MCMC methods, such as the case with only one planet transiting. Specifically, we use a deep learning approach to predict the parameters of non-transit companion for the single transit system with transit information (i.e., TTV, and Transit Duration Variation (TDV)) as input. Thanks to a newly constructed \textit{Transformer}-based architecture that can extract long-range interactions from TTV sequential data, this previously difficult task can now be accomplished with high accuracy, with an overall fractional error of $\sim$2\% on mass and eccentricity.

Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling

We address the outstanding problem of sampling from an unnormalized density that may be non-log-concave and multimodal. To enhance the performance of simple Markov chain Monte Carlo (MCMC) methods, techniques of annealing type have been widely used. However, quantitative theoretical guarantees of these techniques are under-explored. This study takes a first step toward providing a non-asymptotic analysis of annealed MCMC. Specifically, we establish, for the first time, an oracle complexity of $\widetilde{O}\left(\frac{d\beta^2{\cal A}^2}{\varepsilon^6}\right)$ for simple annealed Langevin Monte Carlo algorithm to achieve $\varepsilon^2$ accuracy in Kullback-Leibler divergence to the target distribution $\pi\propto{\rm e}^{-V}$ on $\mathbb{R}^d$ with $\beta$-smooth potential $V$. Here, ${\cal A}$ represents the action of a curve of probability measures interpolating the target distribution $\pi$ and a readily sampleable distribution.

Evaluating the design space of diffusion-based generative models

Most existing theoretical investigations of the accuracy of diffusion models, albeit significant, assume the score function has been approximated to a certain accuracy, and then use this a priori bound to control the error of generation. This article instead provides a first quantitative understanding of the whole generation process, i.e., both training and sampling. More precisely, it conducts a non-asymptotic convergence analysis of denoising score matching under gradient descent. In addition, a refined sampling error analysis for variance exploding models is also provided. The combination of these two results yields a full error analysis, which elucidates (again, but this time theoretically) how to design the training and sampling processes for effective generation. For instance, our theory implies a preference toward noise distribution and loss weighting that qualitatively agree with the ones used in [Karras et al. 2022]. It also provides some perspectives on why the time and variance schedule used in [Karras et al. 2022] could be better tuned than the pioneering version in [Song et al. 2020].

Spectrum-Aware Parameter Efficient Fine-Tuning for Diffusion Models

Adapting large-scale pre-trained generative models in a parameter-efficient manner is gaining traction. Traditional methods like low rank adaptation achieve parameter efficiency by imposing constraints but may not be optimal for tasks requiring high representation capacity. We propose a novel spectrum-aware adaptation framework for generative models. Our method adjusts both singular values and their basis vectors of pretrained weights. Using the Kronecker product and efficient Stiefel optimizers, we achieve parameter-efficient adaptation of orthogonal matrices. We introduce Spectral Orthogonal Decomposition Adaptation (SODA), which balances computational efficiency and representation capacity. Extensive evaluations on text-to-image diffusion models demonstrate SODA's effectiveness, offering a spectrum-aware alternative to existing fine-tuning methods.

Quantitative Convergences of Lie Group Momentum Optimizers

Explicit, momentum-based dynamics that optimize functions defined on Lie groups can be constructed via variational optimization and momentum trivialization. Structure preserving time discretizations can then turn this dynamics into optimization algorithms. This article investigates two types of discretization, Lie Heavy-Ball, which is a known splitting scheme, and Lie NAG-SC, which is newly proposed. Their convergence rates are explicitly quantified under $L$-smoothness and local strong convexity assumptions. Lie NAG-SC provides acceleration over the momentumless case, i.e. Riemannian gradient descent, but Lie Heavy-Ball does not. When compared to existing accelerated optimizers for general manifolds, both Lie Heavy-Ball and Lie NAG-SC are computationally cheaper and easier to implement, thanks to their utilization of group structure. Only gradient oracle and exponential map are required, but not logarithm map or parallel transport which are computational costly.

Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups

The generative modeling of data on manifold is an important task, for which diffusion models in flat spaces typically need nontrivial adaptations. This article demonstrates how a technique called `trivialization' can transfer the effectiveness of diffusion models in Euclidean spaces to Lie groups. In particular, an auxiliary momentum variable was algorithmically introduced to help transport the position variable between data distribution and a fixed, easy-to-sample distribution. Normally, this would incur further difficulty for manifold data because momentum lives in a space that changes with the position. However, our trivialization technique creates to a new momentum variable that stays in a simple $\textbf{fixed vector space}$. This design, together with a manifold preserving integrator, simplifies implementation and avoids inaccuracies created by approximations such as projections to tangent space and manifold, which were typically used in prior work, hence facilitating generation with high-fidelity and efficiency. The resulting method achieves state-of-the-art performance on protein and RNA torsion angle generation and sophisticated torus datasets. We also, arguably for the first time, tackle the generation of data on high-dimensional Special Orthogonal and Unitary groups, the latter essential for quantum problems.

Quantum State Generation with Structure-Preserving Diffusion Model

This article considers the generative modeling of the states of quantum systems, and an approach based on denoising diffusion model is proposed. The key contribution is an algorithmic innovation that respects the physical nature of quantum states. More precisely, the commonly used density matrix representation of mixed-state has to be complex-valued Hermitian, positive semi-definite, and trace one. Generic diffusion models, or other generative methods, may not be able to generate data that strictly satisfy these structural constraints, even if all training data do. To develop a machine learning algorithm that has physics hard-wired in, we leverage the recent development of Mirror Diffusion Model and design a previously unconsidered mirror map, to enable strict structure-preserving generation. Both unconditional generation and conditional generation via classifier-free guidance are experimentally demonstrated efficacious, the latter even enabling the design of new quantum states when generated on unseen labels.

Convergence of Kinetic Langevin Monte Carlo on Lie groups

Explicit, momentum-based dynamics for optimizing functions defined on Lie groups was recently constructed, based on techniques such as variational optimization and left trivialization. We appropriately add tractable noise to the optimization dynamics to turn it into a sampling dynamics, leveraging the advantageous feature that the momentum variable is Euclidean despite that the potential function lives on a manifold. We then propose a Lie-group MCMC sampler, by delicately discretizing the resulting kinetic-Langevin-type sampling dynamics. The Lie group structure is exactly preserved by this discretization. Exponential convergence with explicit convergence rate for both the continuous dynamics and the discrete sampler are then proved under W2 distance. Only compactness of the Lie group and geodesically L-smoothness of the potential function are needed. To the best of our knowledge, this is the first convergence result for kinetic Langevin on curved spaces, and also the first quantitative result that requires no convexity or, at least not explicitly, any common relaxation such as isoperimetry.

Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusion

This paper considers the problem of sampling from non-logconcave distribution, based on queries of its unnormalized density. It first describes a framework, Diffusion Monte Carlo (DMC), based on the simulation of a denoising diffusion process with its score function approximated by a generic Monte Carlo estimator. DMC is an oracle-based meta-algorithm, where its oracle is the assumed access to samples that generate a Monte Carlo score estimator. Then we provide an implementation of this oracle, based on rejection sampling, and this turns DMC into a true algorithm, termed Zeroth-Order Diffusion Monte Carlo (ZOD-MC). We provide convergence analyses by first constructing a general framework, i.e. a performance guarantee for DMC, without assuming the target distribution to be log-concave or satisfying any isoperimetric inequality. Then we prove that ZOD-MC admits an inverse polynomial dependence on the desired sampling accuracy, albeit still suffering from the curse of dimensionality. Consequently, for low dimensional distributions, ZOD-MC is a very efficient sampler, with performance exceeding latest samplers, including also-denoising-diffusion-based RDMC and RS-DMC. Last, we experimentally demonstrate the insensitivity of ZOD-MC to increasingly higher barriers between modes or discontinuity in non-convex potential.

Good regularity creates large learning rate implicit biases: edge of stability, balancing, and catapult

Large learning rates, when applied to gradient descent for nonconvex optimization, yield various implicit biases including the edge of stability (Cohen et al., 2021), balancing (Wang et al., 2022), and catapult (Lewkowycz et al., 2020). These phenomena cannot be well explained by classical optimization theory. Though significant theoretical progress has been made in understanding these implicit biases, it remains unclear for which objective functions would they occur. This paper provides an initial step in answering this question, namely that these implicit biases are in fact various tips of the same iceberg. They occur when the objective function of optimization has some good regularity, which, in combination with a provable preference of large learning rate gradient descent for moving toward flatter regions, results in these nontrivial dynamical phenomena. To establish this result, we develop a new global convergence theory under large learning rates, for a family of nonconvex functions without globally Lipschitz continuous gradient, which was typically assumed in existing convergence analysis. A byproduct is the first non-asymptotic convergence rate bound for large-learning-rate gradient descent optimization of nonconvex functions. We also validate our theory with experiments on neural networks, where different losses, activation functions, and batch normalization all can significantly affect regularity and lead to very different training dynamics.