The latent space of diffusion model mostly still remains unexplored, despite its great success and potential in the field of generative modeling. In fact, the latent space of existing diffusion models are entangled, with a distorted mapping from its latent space to image space. To tackle this problem, we present Isometric Diffusion, equipping a diffusion model with a geometric regularizer to guide the model to learn a geometrically sound latent space of the training data manifold. This approach allows diffusion models to learn a more disentangled latent space, which enables smoother interpolation, more accurate inversion, and more precise control over attributes directly in the latent space. Our extensive experiments consisting of image interpolations, image inversions, and linear editing show the effectiveness of our method.

Deep neural networks (DNNs) continue to grow rapidly in size, making them infeasible to train on a single device. Pipeline parallelism is commonly used in existing DNN systems to support large-scale DNN training by partitioning a DNN into multiple stages, which concurrently perform DNN training for different micro-batches in a pipeline fashion. However, existing pipeline-parallel approaches only consider sequential pipeline stages and thus ignore the topology of a DNN, resulting in missed model-parallel opportunities. This paper presents graph pipeline parallelism (GPP), a new pipeline-parallel scheme that partitions a DNN into pipeline stages whose dependencies are identified by a directed acyclic graph. GPP generalizes existing sequential pipeline parallelism and preserves the inherent topology of a DNN to enable concurrent execution of computationally-independent operators, resulting in reduced memory requirement and improved GPU performance. In addition, we develop GraphPipe, a distributed system that exploits GPP strategies to enable performant and scalable DNN training. GraphPipe partitions a DNN into a graph of stages, optimizes micro-batch schedules for these stages, and parallelizes DNN training using the discovered GPP strategies. Evaluation on a variety of DNNs shows that GraphPipe outperforms existing pipeline-parallel systems such as PipeDream and Piper by up to 1.6X. GraphPipe also reduces the search time by 9-21X compared to PipeDream and Piper.

We explore the top-K rank aggregation problem in which one aims to recover a consistent ordering that focuses on top-K ranked items based on partially revealed preference information. We examine an M-wise comparison model that builds on the Plackett-Luce (PL) model where for each sample, M items are ranked according to their perceived utilities modeled as noisy observations of their underlying true utilities. As our result, we characterize the minimax optimality on the sample size for top-K ranking. The optimal sample size turns out to be inversely proportional to M. We devise an algorithm that effectively converts M-wise samples into pairwise ones and employs a spectral method using the refined data. In demonstrating its optimality, we develop a novel technique for deriving tight $\ell_\infty$ estimation error bounds, which is key to accurately analyzing the performance of top-K ranking algorithms, but has been challenging. Recent work relied on an additional maximum-likelihood estimation (MLE) stage merged with a spectral method to attain good estimates in $\ell_\infty$ error to achieve the limit for the pairwise model. In contrast, although it is valid in slightly restricted regimes, our result demonstrates a spectral method alone to be sufficient for the general M-wise model. We run numerical experiments using synthetic data and confirm that the optimal sample size decreases at the rate of 1/M. Moreover, running our algorithm on real-world data, we find that its applicability extends to settings that may not fit the PL model.

We explore the top-$K$ rank aggregation problem. Suppose a collection of items is compared in pairs repeatedly, and we aim to recover a consistent ordering that focuses on the top-$K$ ranked items based on partially revealed preference information. We investigate the Bradley-Terry-Luce model in which one ranks items according to their perceived utilities modeled as noisy observations of their underlying true utilities. Our main contributions are two-fold. First, in a general comparison model where item pairs to compare are given a priori, we attain an upper and lower bound on the sample size for reliable recovery of the top-$K$ ranked items. Second, more importantly, extending the result to a random comparison model where item pairs to compare are chosen independently with some probability, we show that in slightly restricted regimes, the gap between the derived bounds reduces to a constant factor, hence reveals that a spectral method can achieve the minimax optimality on the (order-wise) sample size required for top-$K$ ranking. That is to say, we demonstrate a spectral method alone to be sufficient to achieve the optimality and advantageous in terms of computational complexity, as it does not require an additional stage of maximum likelihood estimation that a state-of-the-art scheme employs to achieve the optimality. We corroborate our main results by numerical experiments.