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Product of experts (PoE) are layered networks in which the value at each node is an AND (or product) of the values (possibly negated) at its inputs. These were introduced as a neural network architecture that can efficiently learn to generate high-dimensional data which satisfy many low-dimensional constraints---thereby allowing each individual expert to perform a simple task. PoEs have found a variety of applications in learning. We study the problem of identifiability of a product of experts model having a layer of binary latent variables, and a layer of binary observables that are iid conditional on the latents. The previous best upper bound on the number of observables needed to identify the model was exponential in the number of parameters. We show: (a) When the latents are uniformly distributed, the model is identifiable with a number of observables equal to the number of parameters (and hence best possible). (b) In the more general case of arbitrarily distributed latents, the model is identifiable for a number of observables that is still linear in the number of parameters (and within a factor of two of best-possible). The proofs rely on root interlacing phenomena for some special three-term recurrences. 

Abstract Imaging the activity of neurons that are widely distributed across brain regions deep in scattering tissue at high speed remains challenging. Here, we introduce an open-source system with Dual Independent Enhanced Scan Engines for Large field-of-view Two-Photon imaging (Diesel2p). Combining optical design, adaptive optics, and temporal multiplexing, the system offers subcellular resolution over a large field-of-view of ~25 mm 2 , encompassing distances up to 7 mm, with independent scan engines. We demonstrate the flexibility and various use cases of this system for calcium imaging of neurons in the living brain. 

Understanding brain function requires monitoring local and global brain dynamics. Two-photon imaging of the brain across mesoscopic scales has presented trade-offs between imaging area and acquisition speed. We describe a flexible cellular resolution two-photon microscope capable of simultaneous video rate acquisition of four independently targetable brain regions spanning an approximate five-millimeter field of view. With this system, we demonstrate the ability to measure calcium activity across mouse sensorimotor cortex at behaviorally relevant timescales.