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Title: Generative Models for Low-Dimensional Video Representation and Compressive Sensing

Paper content:
Generative priors have become highly effective in solving inverse problems including denoising, inpainting, and reconstruction from few and noisy measurements.
With a generative model we can represent an image with a much lower dimensional latent codes.
In the context of compressive sensing, if the unknown image belongs to the range of a pretrained generative network, then we can recover the image by estimating the underlying compact latent code from the available measurements.
However, recent studies revealed that even untrained deep neural networks can work as a prior for recovering natural images.
These approaches update the network weights keeping latent codes fixed to reconstruct the target image from the given measurements.
In this paper, we optimize over network weights and latent codes to use untrained generative network as prior for video compressive sensing problem.
We show that by optimizing over latent code, we can additionally get concise representation of the frames which retain the structural similarity of the video frames.
We also apply low-rank constraint on the latent codes to represent the video sequences in even lower dimensional latent space.
We empirically show that our proposed methods provide better or comparable accuracy and low computational complexity compared to the existing methods.
Compressive sensing refers to a broad class of problems in which we aim to recover a signal from a small number of measurements [1] - [3] .
Suppose we are given a sequence of measurements for t = 1, . . . , T as y t = A t x t + e t ,
where x t denotes the t th frame in the unknown video sequence, y t denotes its observed measurements, A t denotes the respective measurement operator, and e t denotes noise or error in the measurements.
Our goal is to recover the video sequence (x t ) from the available measurements (y t ).
The recovery problem becomes especially challenging as the number of measurements (in y t ) becomes very small compared to the number of unknowns (in x t ).
Classical signal priors exploit sparse and low-rank structures in images and videos for their reconstruction [4] - [16] .
However, the natural images exhibits far richer nonlinear structures than sparsity alone.
We focus on a newly emerging generative priors that learn a function that maps vectors drawn from a certain distribution in a low-dimensional space into images in a highdimensional space.
The generative model and optimization problems we use are inspired by recent work on using generative models for compressive sensing in [17] - [23] .
Compressive sensing using generative models was introduced in [17] , which used a trained deep generative network as a prior for image reconstruction from compressive measurements.
Afterwards deep image prior (DIP) used an untrained convolutional generative model as a prior for solving inverse problems such as inpainting and denoising because of their tendency to generate natural images [22] ; the reconstruction problem involves optimization of generator network parameters.
Inspired by these observations, a number of methods have been proposed for solving compressive sensing problem by optimizing generator network weights while keeping the latent code fixed at a random value [19] , [20] .
Both DIP [22] and deep decoder [20] update the model weights to generate a given image; therefore, the generator can reconstruct wide range of images.
One key difference between the two approaches is that the network used in DIP is highly overparameterized, while the one used in deep decoder is underparameterized.
We observed two main limitations in the DIP and deep decoder-based video recovery that we seek to address in this paper.
(1) The latent codes in DIP and deep decoder methods are initialized at random and stay fixed throughout the recovery process.
Therefore, we cannot infer the structural similarities in the images from the structural similarities in the latent codes.
(2) Both of these methods train one network per image.
A naive approach to train one network per frame in a video will be computationally prohibitive, and if we train a single network to generate the entire video sequence, then their performance degrades.
Therefore, we propose joint optimization over network weights γ and the latent codes z t to reconstruct video sequence.
Thus we learn a single generator and a set of latent codes to represent a video sequence.
We observe that when we optimize over latent code alongside network weights, the temporal similarity in the video frames is reflected in the latent code representation.
To exploit similarities among the frames in a video sequence, we also include low-rank constraints on the latent codes.
An illustration of different types of representations we use in this paper are shown in Figure 1 .

assistant: text: Recover videos from compressive measurements by learning a low-dimensional (low-rank) representation directly from measurements while training a deep generator.