Sum-of-squares lower bounds for Non-Gaussian Component Analysis

Non-Gaussian Component Analysis (NGCA) is the statistical task of finding a non-Gaussian direction in a high-dimensional dataset. Specifically, given i.i.d.\ samples from a distribution $P^A_{v}$ on $\mathbb{R}^n$ that behaves like a known distribution $A$ in a hidden direction $v$ and like a standard Gaussian in the orthogonal complement, the goal is to approximate the hidden direction. The standard formulation posits that the first $k-1$ moments of $A$ match those of the standard Gaussian and the $k$-th moment differs. Under mild assumptions, this problem has sample complexity $O(n)$. On the other hand, all known efficient algorithms require $\Omega(n^{k/2})$ samples. Prior work developed sharp Statistical Query and low-degree testing lower bounds suggesting an information-computation tradeoff for this problem. Here we study the complexity of NGCA in the Sum-of-Squares (SoS) framework. Our main contribution is the first super-constant degree SoS lower bound for NGCA. Specifically, we show that if the non-Gaussian distribution $A$ matches the first $(k-1)$ moments of $\mathcal{N}(0, 1)$ and satisfies other mild conditions, then with fewer than $n^{(1 - \varepsilon)k/2}$ many samples from the normal distribution, with high probability, degree $(\log n)^{{1\over 2}-o_n(1)}$ SoS fails to refute the existence of such a direction $v$. Our result significantly strengthens prior work by establishing a super-polynomial information-computation tradeoff against a broader family of algorithms. As corollaries, we obtain SoS lower bounds for several problems in robust statistics and the learning of mixture models. Our SoS lower bound proof introduces a novel technique, that we believe may be of broader interest, and a number of refinements over existing methods.

An inversely designed mode-multiplexed photonic integrated vector dot-product core

Photonic computing has the potential of harnessing the full degrees of freedom (DOFs) of the light field, including wavelength, spatial mode, spatial location, phase quadrature, and polarization, to achieve higher level of computation parallelization and scalability than digital electronic processors. While multiplexing using wavelength and other DOFs can be readily integrated on silicon photonics platforms with compact footprints, conventional mode-division multiplexed (MDM) photonic designs occupy areas exceeding tens to hundreds of microns for a few spatial modes, significantly limiting their scalability. Here we utilize inverse design to demonstrate an ultracompact photonic computing core that calculates vector dot-products based on MDM coherent mixing within a nominal footprint of 5 um x 3 um. Our dot-product core integrates the functionalities of 2 mode multiplexers and 1 multi-mode coherent mixers, all within the footprint, and could be applied to various computation and computer vision tasks, with high computing throughput density. We experimentally demonstrate computing examples on the fabricated core, including complex number multiplication and motion estimation using optical flow.

Integrated Photonic Encoder for Terapixel Image Processing

Modern lens designs are capable of resolving >10 gigapixels, while advances in camera frame-rate and hyperspectral imaging have made Terapixel/s data acquisition a real possibility. The main bottlenecks preventing such high data-rate systems are power consumption and data storage. In this work, we show that analog photonic encoders could address this challenge, enabling high-speed image compression using orders-of-magnitude lower power than digital electronics. Our approach relies on a silicon-photonics front-end to compress raw image data, foregoing energy-intensive image conditioning and reducing data storage requirements. The compression scheme uses a passive disordered photonic structure to perform kernel-type random projections of the raw image data with minimal power consumption and low latency. A back-end neural network can then reconstruct the original images with structural similarity exceeding 90%. This scheme has the potential to process Terapixel/s data streams using less than 100 fJ/pixel, providing a path to ultra-high-resolution data and image acquisition systems.

A physical neural network training approach toward multi-plane light conversion design

Multi-plane light converter (MPLC) designs supporting hundreds of modes are attractive in high-throughput optical communications. These photonic structures typically comprise >10 phase masks in free space, with millions of independent design parameters. Conventional MPLC design using wavefront matching updates one mask at a time while fixing the rest. Here we construct a physical neural network (PNN) to model the light propagation and phase modulation in MPLC, providing access to the entire parameter set for optimization, including not only profiles of the phase masks and the distances between them. PNN training supports flexible optimization sequences and is a superset of existing MPLC design methods. In addition, our method allows tuning of hyperparameters of PNN training such as learning rate and batch size. Because PNN-based MPLC is found to be insensitive to the number of input and target modes in each training step, we have demonstrated a high-order MPLC design (45 modes) using mini batches that fit into the available computing resources.

Training of mixed-signal optical convolutional neural network with reduced quantization level

Mixed-signal artificial neural networks (ANNs) that employ analog matrix-multiplication accelerators can achieve higher speed and improved power efficiency. Though analog computing is known to be susceptible to noise and device imperfections, various analog computing paradigms have been considered as promising solutions to address the growing computing demand in machine learning applications, thanks to the robustness of ANNs. This robustness has been explored in low-precision, fixed-point ANN models, which have proven successful on compressing ANN model size on digital computers. However, these promising results and network training algorithms cannot be easily migrated to analog accelerators. The reason is that digital computers typically carry intermediate results with higher bit width, though the inputs and weights of each ANN layers are of low bit width; while the analog intermediate results have low precision, analogous to digital signals with a reduced quantization level. Here we report a training method for mixed-signal ANN with two types of errors in its analog signals, random noise, and deterministic errors (distortions). The results showed that mixed-signal ANNs trained with our proposed method can achieve an equivalent classification accuracy with noise level up to 50% of the ideal quantization step size. We have demonstrated this training method on a mixed-signal optical convolutional neural network based on diffractive optics.

Signal retrieval with measurement system knowledge using variational generative model

Signal retrieval from a series of indirect measurements is a common task in many imaging, metrology and characterization platforms in science and engineering. Because most of the indirect measurement processes are well-described by physical models, signal retrieval can be solved with an iterative optimization that enforces measurement consistency and prior knowledge on the signal. These iterative processes are time-consuming and only accommodate a linear measurement process and convex signal constraints. Recently, neural networks have been widely adopted to supersede iterative signal retrieval methods by approximating the inverse mapping of the measurement model. However, networks with deterministic processes have failed to distinguish signal ambiguities in an ill-posed measurement system, and retrieved signals often lack consistency with the measurement. In this work we introduce a variational generative model to capture the distribution of all possible signals, given a particular measurement. By exploiting the known measurement model in the variational generative framework, our signal retrieval process resolves the ambiguity in the forward process, and learns to retrieve signals that satisfy the measurement with high fidelity in a variety of linear and nonlinear ill-posed systems, including ultrafast pulse retrieval, coded aperture compressive video sensing and image retrieval from Fresnel hologram.