Henry Li

Uri Shaham, Kelly Stanton, Henry Li et al.

Spectral clustering is a leading and popular technique in unsupervised data analysis. Two of its major limitations are scalability and generalization of the spectral embedding (i.e., out-of-sample-extension). In this paper we introduce a deep learning approach to spectral clustering that overcomes the above shortcomings. Our network, which we call SpectralNet, learns a map that embeds input data points into the eigenspace of their associated graph Laplacian matrix and subsequently clusters them. We train SpectralNet using a procedure that involves constrained stochastic optimization. Stochastic optimization allows it to scale to large datasets, while the constraints, which are implemented using a special-purpose output layer, allow us to keep the network output orthogonal. Moreover, the map learned by SpectralNet naturally generalizes the spectral embedding to unseen data points. To further improve the quality of the clustering, we replace the standard pairwise Gaussian affinities with affinities leaned from unlabeled data using a Siamese network. Additional improvement can be achieved by applying the network to code representations produced, e.g., by standard autoencoders. Our end-to-end learning procedure is fully unsupervised. In addition, we apply VC dimension theory to derive a lower bound on the size of SpectralNet. State-of-the-art clustering results are reported on the Reuters dataset. Our implementation is publicly available at https://github.com/kstant0725/SpectralNet .

Henry Li, Marcus Pereira

Existing approaches to diffusion-based inverse problem solvers frame the signal recovery task as a probabilistic sampling episode, where the solution is drawn from the desired posterior distribution. This framework suffers from several critical drawbacks, including the intractability of the conditional likelihood function, strict dependence on the score network approximation, and poor $\mathbf{x}_0$ prediction quality. We demonstrate that these limitations can be sidestepped by reframing the generative process as a discrete optimal control episode. We derive a diffusion-based optimal controller inspired by the iterative Linear Quadratic Regulator (iLQR) algorithm. This framework is fully general and able to handle any differentiable forward measurement operator, including super-resolution, inpainting, Gaussian deblurring, nonlinear deblurring, and even highly nonlinear neural classifiers. Furthermore, we show that the idealized posterior sampling equation can be recovered as a special case of our algorithm. We then evaluate our method against a selection of neural inverse problem solvers, and establish a new baseline in image reconstruction with inverse problems.

Hannah Lawrence, David A. Barmherzig, Henry Li et al.

Phase retrieval is the inverse problem of recovering a signal from magnitude-only Fourier measurements, and underlies numerous imaging modalities, such as Coherent Diffraction Imaging (CDI). A variant of this setup, known as holography, includes a reference object that is placed adjacent to the specimen of interest before measurements are collected. The resulting inverse problem, known as holographic phase retrieval, is well-known to have improved problem conditioning relative to the original. This innovation, i.e. Holographic CDI, becomes crucial at the nanoscale, where imaging specimens such as viruses, proteins, and crystals require low-photon measurements. This data is highly corrupted by Poisson shot noise, and often lacks low-frequency content as well. In this work, we introduce a dataset-free deep learning framework for holographic phase retrieval adapted to these challenges. The key ingredients of our approach are the explicit and flexible incorporation of the physical forward model into an automatic differentiation procedure, the Poisson log-likelihood objective function, and an optional untrained deep image prior. We perform extensive evaluation under realistic conditions. Compared to competing classical methods, our method recovers signal from higher noise levels and is more resilient to suboptimal reference design, as well as to large missing regions of low frequencies in the observations. Finally, we show that these properties carry over to experimental data acquired on optical wavelengths. To the best of our knowledge, this is the first work to consider a dataset-free machine learning approach for holographic phase retrieval.

Henry Li, Ofir Lindenbaum, Xiuyuan Cheng et al.

Variational autoencoders (VAEs) and generative adversarial networks (GANs) enjoy an intuitive connection to manifold learning: in training the decoder/generator is optimized to approximate a homeomorphism between the data distribution and the sampling space. This is a construction that strives to define the data manifold. A major obstacle to VAEs and GANs, however, is choosing a suitable prior that matches the data topology. Well-known consequences of poorly picked priors are posterior and mode collapse. To our knowledge, no existing method sidesteps this user choice. Conversely, $\textit{diffusion maps}$ automatically infer the data topology and enjoy a rigorous connection to manifold learning, but do not scale easily or provide the inverse homeomorphism (i.e. decoder/generator). We propose a method that combines these approaches into a generative model that inherits the asymptotic guarantees of $\textit{diffusion maps}$ while preserving the scalability of deep models. We prove approximation theoretic results for the dimension dependence of our proposed method. Finally, we demonstrate the effectiveness of our method with various real and synthetic datasets.