Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models

Diffusion models benefit from instillation of task-specific information into the score function to steer the sample generation towards desired properties. Such information is coined as guidance. For example, in text-to-image synthesis, text input is encoded as guidance to generate semantically aligned images. Proper guidance inputs are closely tied to the performance of diffusion models. A common observation is that strong guidance promotes a tight alignment to the task-specific information, while reducing the diversity of the generated samples. In this paper, we provide the first theoretical study towards understanding the influence of guidance on diffusion models in the context of Gaussian mixture models. Under mild conditions, we prove that incorporating diffusion guidance not only boosts classification confidence but also diminishes distribution diversity, leading to a reduction in the differential entropy of the output distribution. Our analysis covers the widely adopted sampling schemes including DDPM and DDIM, and leverages comparison inequalities for differential equations as well as the Fokker-Planck equation that characterizes the evolution of probability density function, which may be of independent theoretical interest.

Failures and Successes of Cross-Validation for Early-Stopped Gradient Descent

We analyze the statistical properties of generalized cross-validation (GCV) and leave-one-out cross-validation (LOOCV) applied to early-stopped gradient descent (GD) in high-dimensional least squares regression. We prove that GCV is generically inconsistent as an estimator of the prediction risk of early-stopped GD, even for a well-specified linear model with isotropic features. In contrast, we show that LOOCV converges uniformly along the GD trajectory to the prediction risk. Our theory requires only mild assumptions on the data distribution and does not require the underlying regression function to be linear. Furthermore, by leveraging the individual LOOCV errors, we construct consistent estimators for the entire prediction error distribution along the GD trajectory and consistent estimators for a wide class of error functionals. This in particular enables the construction of pathwise prediction intervals based on GD iterates that have asymptotically correct nominal coverage conditional on the training data.

Sharp Analysis of Power Iteration for Tensor PCA

We investigate the power iteration algorithm for the tensor PCA model introduced in Richard and Montanari (2014). Previous work studying the properties of tensor power iteration is either limited to a constant number of iterations, or requires a non-trivial data-independent initialization. In this paper, we move beyond these limitations and analyze the dynamics of randomly initialized tensor power iteration up to polynomially many steps. Our contributions are threefold: First, we establish sharp bounds on the number of iterations required for power method to converge to the planted signal, for a broad range of the signal-to-noise ratios. Second, our analysis reveals that the actual algorithmic threshold for power iteration is smaller than the one conjectured in literature by a polylog(n) factor, where n is the ambient dimension. Finally, we propose a simple and effective stopping criterion for power iteration, which provably outputs a solution that is highly correlated with the true signal. Extensive numerical experiments verify our theoretical results.

Deep Networks as Denoising Algorithms: Sample-Efficient Learning of Diffusion Models in High-Dimensional Graphical Models

We investigate the approximation efficiency of score functions by deep neural networks in diffusion-based generative modeling. While existing approximation theories utilize the smoothness of score functions, they suffer from the curse of dimensionality for intrinsically high-dimensional data. This limitation is pronounced in graphical models such as Markov random fields, common for image distributions, where the approximation efficiency of score functions remains unestablished. To address this, we observe score functions can often be well-approximated in graphical models through variational inference denoising algorithms. Furthermore, these algorithms are amenable to efficient neural network representation. We demonstrate this in examples of graphical models, including Ising models, conditional Ising models, restricted Boltzmann machines, and sparse encoding models. Combined with off-the-shelf discretization error bounds for diffusion-based sampling, we provide an efficient sample complexity bound for diffusion-based generative modeling when the score function is learned by deep neural networks.

ImageReward: Learning and Evaluating Human Preferences for Text-to-Image Generation

We present ImageReward -- the first general-purpose text-to-image human preference reward model -- to address various prevalent issues in generative models and align them with human values and preferences. Its training is based on our systematic annotation pipeline that covers both the rating and ranking components, collecting a dataset of 137k expert comparisons to date. In human evaluation, ImageReward outperforms existing scoring methods (e.g., CLIP by 38.6\%), making it a promising automatic metric for evaluating and improving text-to-image synthesis. The reward model is publicly available via the \texttt{image-reward} package at \url{https://github.com/THUDM/ImageReward}.

Gestalt-Guided Image Understanding for Few-Shot Learning

Due to the scarcity of available data, deep learning does not perform well on few-shot learning tasks. However, human can quickly learn the feature of a new category from very few samples. Nevertheless, previous work has rarely considered how to mimic human cognitive behavior and apply it to few-shot learning. This paper introduces Gestalt psychology to few-shot learning and proposes Gestalt-Guided Image Understanding, a plug-and-play method called GGIU. Referring to the principle of totality and the law of closure in Gestalt psychology, we design Totality-Guided Image Understanding and Closure-Guided Image Understanding to extract image features. After that, a feature estimation module is used to estimate the accurate features of images. Extensive experiments demonstrate that our method can improve the performance of existing models effectively and flexibly without retraining or fine-tuning. Our code is released on https://github.com/skingorz/GGIU.

Distribution Aligned Feature Clustering for Zero-Shot Sketch-Based Image Retrieval

Zero-Shot Sketch-Based Image Retrieval (ZS-SBIR) is a challenging cross-modal retrieval task. In prior arts, the retrieval is conducted by sorting the distance between the query sketch and each image in the gallery. However, the domain gap and the zero-shot setting make neural networks hard to generalize. This paper tackles the challenges from a new perspective: utilizing gallery image features. We propose a Cluster-then-Retrieve (ClusterRetri) method that performs clustering on the gallery images and uses the cluster centroids as proxies for retrieval. Furthermore, a distribution alignment loss is proposed to align the image and sketch features with a common Gaussian distribution, reducing the domain gap. Despite its simplicity, our proposed method outperforms the state-of-the-art methods by a large margin on popular datasets, e.g., up to 31% and 39% relative improvement of mAP@all on the Sketchy and TU-Berlin datasets.

Lower Bounds for the Convergence of Tensor Power Iteration on Random Overcomplete Models

Tensor decomposition serves as a powerful primitive in statistics and machine learning. In this paper, we focus on using power iteration to decompose an overcomplete random tensor. Past work studying the properties of tensor power iteration either requires a non-trivial data-independent initialization, or is restricted to the undercomplete regime. Moreover, several papers implicitly suggest that logarithmically many iterations (in terms of the input dimension) are sufficient for the power method to recover one of the tensor components. In this paper, we analyze the dynamics of tensor power iteration from random initialization in the overcomplete regime. Surprisingly, we show that polynomially many steps are necessary for convergence of tensor power iteration to any of the true component, which refutes the previous conjecture. On the other hand, our numerical experiments suggest that tensor power iteration successfully recovers tensor components for a broad range of parameters, despite that it takes at least polynomially many steps to converge. To further complement our empirical evidence, we prove that a popular objective function for tensor decomposition is strictly increasing along the power iteration path. Our proof is based on the Gaussian conditioning technique, which has been applied to analyze the approximate message passing (AMP) algorithm. The major ingredient of our argument is a conditioning lemma that allows us to generalize AMP-type analysis to non-proportional limit and polynomially many iterations of the power method.

Along Similar Lines: Local Obstacle Avoidance for Long-term Autonomous Path Following

Visual Teach and Repeat 3 (VT&R3), a generalization of stereo VT&R, achieves long-term autonomous path-following using topometric mapping and localization from a single rich sensor stream. In this paper, we improve the capabilities of a LiDAR implementation of VT&R3 to reliably detect and avoid obstacles in changing environments. Our architecture simplifies the obstacle-perception problem to that of place-dependent change detection. We then extend the behaviour of generic sample-based motion planners to better suit the teach-and-repeat problem structure by introducing a new edge-cost metric paired with a curvilinear planning space. The resulting planner generates naturally smooth paths that avoid local obstacles while minimizing lateral path deviation to best exploit prior terrain knowledge. While we use the method with VT&R, it can be generalized to suit arbitrary path-following applications. Experimental results from online run-time analysis, unit testing, and qualitative experiments on a differential drive robot show the promise of the technique for reliable long-term autonomous operation in complex unstructured environments.

Picking Up Speed: Continuous-Time Lidar-Only Odometry using Doppler Velocity Measurements

Frequency-Modulated Continuous-Wave (FMCW) lidar is a recently emerging technology that additionally enables per-return instantaneous relative radial velocity measurements via the Doppler effect. In this letter, we present the first continuous-time lidar-only odometry algorithm using these Doppler velocity measurements from an FMCW lidar to aid odometry in geometrically degenerate environments. We apply an existing continuous-time framework that efficiently estimates the vehicle trajectory using Gaussian process regression to compensate for motion distortion due to the scanning-while-moving nature of any mechanically actuated lidar (FMCW and non-FMCW). We evaluate our proposed algorithm on several real-world datasets, including publicly available ones and datasets we collected. Our algorithm outperforms the only existing method that also uses Doppler velocity measurements, and we study difficult conditions where including this extra information greatly improves performance. We additionally demonstrate state-of-the-art performance of lidar-only odometry with and without using Doppler velocity measurements in nominal conditions. Code for this project can be found at: https://github.com/utiasASRL/steam_icp.