Dimensionality Reduction for Stationary Time Series via Stochastic Nonconvex Optimization

Stochastic optimization naturally arises in machine learning. Efficient algorithms with provable guarantees, however, are still largely missing, when the objective function is nonconvex and the data points are dependent. This paper studies this fundamental challenge through a streaming PCA problem for stationary time series data. Specifically, our goal is to estimate the principle component of time series data with respect to the covariance matrix of the stationary distribution. Computationally, we propose a variant of Oja's algorithm combined with downsampling to control the bias of the stochastic gradient caused by the data dependency. Theoretically, we quantify the uncertainty of our proposed stochastic algorithm based on diffusion approximations. This allows us to prove the asymptotic rate of convergence and further implies near optimal asymptotic sample complexity. Numerical experiments are provided to support our analysis

Observation of photon-phonon correlations via dissipative filtering

Cavity-optomechanics enables photon-phonon interaction and correlations by harnessing the radiation-pressure force. Here, we realize a ``cavity-in-a-membrane'' optomechanical architecture which allows detection of the motion of lithographically-defined, ultrathin membranes via an integrated optical cavity. Using a dissipative filtering method, we are able to eliminate the probe light in situ and observe photon-phonon correlations associated with the low-frequency membrane mechanical mode. The developed method is generally applicable for study of low-frequency light scattering processes where conventional frequency-selective filtering is unfeasible

A rigid, low-loss fiber-optic coupler for cryogenic photonics

Recent developments in quantum light-matter coupled systems and quantum transducers have highlighted the need for cryogenic optical measurements. In this study, we present a mechanically-rigid fiber-optic coupler with a coupling efficiency of over 50% for telecom wavelength light at cryogenic temperatures. Our method enables sensitive photonic device measurements that are alignment-free and immune to mechanical vibrations in cryogenic setups

Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data

Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Furthermore, the generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution. The convergence rate depends on the subspace dimension, indicating that diffusion models can circumvent the curse of data ambient dimensionality.Comment: 52 pages, 4 figure

Quantum correlated photons via a passive nonlinear microcavity

Photons, by nature, typically do not exhibit interactions with each other. Creating photon-photon interactions holds immense importance in both fundamental physics and quantum technologies. Currently, such interactions have only been achieved indirectly as mediated by atomic-like quantum emitters with resonant photon-atom interactions. However, the use of these indirect interactions presents substantial fundamental challenges that impede scaling and practical applications. Here we demonstrate creation of non-classical photon correlations, including photon anti-bunching, via a passive InGaP photonic integrated circuit. Our approach employs the quantum interference between uncorrelated light and the two-photon bound state, the latter of which arises from the $\chi^{(2)}$-mediated photon interaction. Our work opens a new route in controlling quantum light by harnessing highly-engineerable bulk optical nonlinearities, which has significant implications for nonlinear optical quantum information processing and quantum networking.Comment: 26 pages, 15 figures, 2 table