Non-convex Sparse Deviation Modeling Via Generative Models

In the information age,people's demand and quality for image,audio,video and other data signals are getting higher and higher.Traditional sampling methods face many difficulties in data sampling,compression,processing,storage and so on.Compressed sensing is proposed as a new theory for high-dimensional data process-ing.It can accurately reconstruct the signal from a small number of linear measure-ments by using the sparsity or compressibility of the signal.In recent years,many researchers have integrated deep learning into compressed sensing and carried out a large number of experiments,especially in generating models.A domain-specific generative model can provide a stronger structural priori,which allows the original signal to be recovered with fewer measurements.This method no longer restricts the signal to be sparse,but directly extracts the structural hypothesis of the signal from the data features.However,in order to overcome the problem that this method can only guarantee accurate recovery on its support set,a "Sparse-Gen" framework is proposed.In this paper,on the basis of sparse deviation modeling based on generation model,a non-convex sparse deviation model is proposed,which reduces the measurement condition for successful restoration and improves the restoration effect.The main contents of this paper are as follows:1.The background and research status of compressed sensing theory are sum-marized,and three parts of compressed sensing theory(sparse representation of signal,construction of measurement matrix and signal reconstruction)and two gen-erative models based on deep learning(Generative Adversarial Networks and Vari-ational Auto-Encoder)are introduced in detail.2.Under the compressed sensing theory,the non-convex sparse deviation model based on generation model(Lq-Gen,0<q?1)is studied,and the corresponding theoretical results are given.For(Lq-Gen),we propose the restricted isometric prop-erty(q-RIP)and the restricted eigenvalue condition of the set(q-S-REC)based on the Lq norm.When the recovery signal is within the non-convex sparse deviation range of the generator,if the measurement matrix satisfies the above two condition-s,the reconstruction error of the optimal decoding has an upper bound,and this upper bound is closely related to the value of q.Then,the measurement number conditions for successful recovery with high probability under the condition of gen-erating function are derived.The first term is mainly related to the logarithm of the Lipschitz constant of the model function.The larger the constant L is,the more difficult it is to recover.The decrease of q allows fewer measurements to recover the signal successfully.With the dependence on n vanishing as q? 0.Moreover,all the conclusions can be reduced to the results under the original Sparse-Gen framework at q=1.3.In order to verify the effectiveness and superiority of the Lq-Gen proposed in this paper,we have carried out a series of experiments using three image data sets and two generated model types.For MNIST and F-MNIST datasets,Lq-Gen based on VAE is used for recovery,while CeleA datasets are reconstructed using DCGAN's Lq-Gen model.Under the condition of minimal measurement number,the reconstruction error of our method is obviously better than that of LASSO-based recovery,generation model-based recovery and Sparse-Gen,especially when q=0.5.Finally,in terms of noise tolerance,Lq-Gen algorithm has stronger anti-noise ability than LASSO algorithm.