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Title: NOODL: Provable Online Dictionary Learning and Sparse Coding

Paper content:
We consider the dictionary learning problem, where the aim is to model the given data as a linear combination of a few columns of a matrix known as a dictionary, where the sparse weights forming the linear combination are known as coefficients.
Since the dictionary and coefficients, parameterizing the linear model are unknown, the corresponding optimization is inherently non-convex.
This was a major challenge until recently, when provable algorithms for dictionary learning were proposed.
Yet, these provide guarantees only on the recovery of the dictionary, without explicit recovery guarantees on the coefficients.
Moreover, any estimation error in the dictionary adversely impacts the ability to successfully localize and estimate the coefficients.
This potentially limits the utility of existing provable dictionary learning methods in applications where coefficient recovery is of interest.
To this end, we develop NOODL: a simple Neurally plausible alternating Optimization-based Online Dictionary Learning algorithm, which recovers both the dictionary and coefficients exactly at a geometric rate, when initialized appropriately.
Our algorithm, NOODL, is also scalable and amenable for large scale distributed implementations in neural architectures, by which we mean that it only involves simple linear and non-linear operations.
Finally, we corroborate these theoretical results via experimental evaluation of the proposed algorithm with the current state-of-the-art techniques.
Sparse models avoid overfitting by favoring simple yet highly expressive representations.
Since signals of interest may not be inherently sparse, expressing them as a sparse linear combination of a few columns of a dictionary is used to exploit the sparsity properties.
Of specific interest are overcomplete dictionaries, since they provide a flexible way of capturing the richness of a dataset, while yielding sparse representations that are robust to noise; see BID13 ; Chen et al. (1998); Donoho et al. (2006) .
In practice however, these dictionaries may not be known, warranting a need to learn such representations -known as dictionary learning (DL) or sparse coding BID14 .
Formally, this entails learning an a priori unknown dictionary A ∈ R n×m and sparse coefficients x * (j) ∈ R m from data samples y (j) ∈ R n generated as DISPLAYFORM0 This particular model can also be viewed as an extension of the low-rank model BID15 .
Here, instead of sharing a low-dimensional structure, each data vector can now reside in a separate low-dimensional subspace.
Therefore, together the data matrix admits a union-of-subspace model.
As a result of this additional flexibility, DL finds applications in a wide range of signal processing and machine learning tasks, such as denoising (Elad and Aharon, 2006) , image inpainting BID12 , clustering and classification (Ramirez et al., 2010; BID16 BID17 BID18 2019b; a) , and analysis of deep learning primitives (Ranzato et al., 2008; BID0 ; see also Elad (2010) , and references therein.Notwithstanding the non-convexity of the associated optimization problems (since both factors are unknown), alternating minimization-based dictionary learning techniques have enjoyed significant success in practice.
Popular heuristics include regularized least squares-based BID14 BID8 BID12 BID9 BID7 , and greedy approaches such as the method of optimal directions (MOD) (Engan et al., 1999) and k-SVD (Aharon et al., 2006) .
However, dictionary learning, and matrix factorization models in general, are difficult to analyze in theory; see also BID10 .To
this end, motivated from a string of recent theoretical works BID1 BID4 Geng and Wright, 2014) , provable algorithms for DL have been proposed recently to explain the success of aforementioned alternating minimization-based algorithms (Agarwal et al., 2014; Arora et al., 2014; BID20 . However
, these works exclusively focus on guarantees for dictionary recovery. On the
other hand, for applications of DL in tasks such as classification and clusteringwhich rely on coefficient recovery -it is crucial to have guarantees on coefficients recovery as well.Contrary to conventional prescription, a sparse approximation step after recovery of the dictionary does not help; since any error in the dictionary -which leads to an error-in-variables (EIV) (Fuller, 2009 ) model for the dictionary -degrades our ability to even recover the support of the coefficients (Wainwright, 2009) . Further
, when this error is non-negligible, the existing results guarantee recovery of the sparse coefficients only in 2 -norm sense (Donoho et al., 2006) . As a result
, there is a need for scalable dictionary learning techniques with guaranteed recovery of both factors.
we note that Arora15(''biased'') and Arora15(''unbiased'') incur significant bias, while NOODL converges to A * linearly.
NOODL also converges for significantly higher choices of sparsity k, i.e., for k = 100 as shown in panel (d), beyond k = O( √ n), indicating a potential for improving this bound.
Further, we observe that Mairal '09 exhibits significantly slow convergence as compared to NOODL.
Also, in panels (a-ii), (b-ii), (c-ii) and (d-ii) we show the corresponding performance of NOODL in terms of the error in the overall fit ( Y − AX F / Y F ), and the error in the coefficients and the dictionary, in terms of relative Frobenius error metric discussed above.
We observe that the error in dictionary and coefficients drops linearly as indicated by our main result.
We present NOODL, to the best of our knowledge, the first neurally plausible provable online algorithm for exact recovery of both factors of the dictionary learning (DL) model.
NOODL alternates between:
(a) an iterative hard thresholding (IHT)-based step for coefficient recovery, and
(b) a gradient descent-based update for the dictionary, resulting in a simple and scalable algorithm, suitable for large-scale distributed implementations.
We show that once initialized appropriately, the sequence of estimates produced by NOODL converge linearly to the true dictionary and coefficients without incurring any bias in the estimation.
Complementary to our theoretical and numerical results, we also design an implementation of NOODL in a neural architecture for use in practical applications.
In essence, the analysis of this inherently non-convex problem impacts other matrix and tensor factorization tasks arising in signal processing, collaborative filtering, and machine learning.

assistant: text: We present a provable algorithm for exactly recovering both factors of the dictionary learning model.