Matrix Completion via Nonsmooth Regularization of Fully Connected Neural Networks

arXiv:2403.10232 · cs.IT, cs.LG, math.IT · Submitted 2026-08-10 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Next we'll be talking about the paper "Matrix Completion via Nonsmooth Regularization of Fully Connected Neural Networks".

Jane: The paper was written by Sajad Faramarzi, Farzan Haddadi, Sajjad Amini, Masoud Ahookhosh and Symeon Chatzinotas from Iran University of Science & Technology and Sharif University of Technology and University of Massachusetts Amherst and University of Antwerp and University of Luxembourg.

Tom: Stay tuned as we take you through the paper and discuss its implications.

Title: Tom: Welcome back to the show, everyone. Today we're digging into a paper that's got a real mouthful of a title: "Matrix Completion via Nonsmooth Regularization of Fully Connected Neural Networks."

Jane: And I'm Jane. Tom, before we even get to the math, let's just unpack that title, because it's a lot. Matrix completion is basically the problem of filling in missing pieces of a table of data.

Tom: Right, like if you have a spreadsheet of movie ratings and half the cells are empty, you want to guess what's in the blanks.

Jane: Exactly. And the "fully connected neural networks" part is the tool they're using to do the guessing. It's a type of AI model where every neuron in one layer talks to every neuron in the next.

Tom: So the title is basically saying, "We're using a big AI brain to fill in the blanks." But what about "nonsmooth regularization"? That sounds like a mouthful.

Jane: It is. Regularization is a trick to stop the AI from memorizing the data it sees and instead learning the general pattern. "Nonsmooth" just means the math they're using has some sharp corners in it.

Tom: Sharp corners. Like a jagged line instead of a smooth curve. And that's a problem because most training methods need smooth curves to work.

Jane: Right. So the title is telling you they found a way to use this sharp-cornered math to keep the AI from overfitting, which is when it gets too good at the training data and bad at everything else.

Lu: And that's the real story here. I'm Lu, by the way. The authors are tackling the core problem of overfitting in these deep networks, which is huge when you have very little data to work with.

Meng: I'm Meng. And from an engineering standpoint, overfitting is the thing that kills you in practice. You train a model, it looks great, and then it falls apart on real-world data.

Tom: So this paper is about making these AI models actually reliable when the data is sparse and messy?

Jane: Exactly. And the fact that they're using "nonsmooth" math is the clever part, because it's usually avoided. It's like they found a way to use a tool everyone else was scared to touch.

Lu: And they're showing it works better than the standard approaches. That's the exciting part.

Tom: Well, I'm hooked. Let's get into the nitty-gritty of what they actually did.

Summary: Jane: So, Tom, we've got the title figured out. Now let's talk about what the paper actually accomplishes. The summary is pretty bold.

Tom: It is. They're claiming their method, which they call DNN-NSR, beats out a bunch of existing algorithms on synthetic data, image inpainting, and even movie recommendation systems.

Jane: And the key to that success, as we said, is controlling overfitting. But how? They're using two specific types of regularization.

Tom: Right, the first is the l1 norm on the intermediate layers of the network. That's a fancy way of saying they're encouraging the network to use fewer active connections, making it more sparse.

Jane: And the second is the nuclear norm on the weight matrices. That's a way to encourage the weight matrices to be low-rank, which is a fancy way of saying they want the network to be simpler and more efficient.

Lu: And the trick is, these regularizers are nonsmooth. They have those sharp corners we talked about. That makes the math harder, but it also makes the regularization much stronger.

Meng: Which is why they had to invent a new training algorithm. You can't just use standard gradient descent when your loss function has sharp corners.

Tom: Right, so they propose a variant of the proximal gradient method. It's a way to handle those nonsmooth parts.

Jane: And they prove that their algorithm actually converges to a good solution. They don't just show it works in practice; they show it mathematically.

Lu: That's the rigorous part that makes this a solid contribution. They're not just throwing a bunch of regularizers at the problem and hoping it works.

Meng: And the results back it up. On the synthetic data, they see a significant jump in PSNR, which is a measure of reconstruction quality. For example, at an eighty percent missing rate, they're getting a couple of decibels better than the next best method.

Tom: So they're filling in more blanks more accurately, even when almost all the data is gone.

Jane: And that's the dream, right? Making these models work when you have very little to go on.

Lu: It's a strong summary. They've identified a real problem, proposed a clever solution, and backed it up with both theory and experiments.

Tom: Okay, so the summary is impressive. But what's the actual improvement they're proposing over previous work? Let's dig into that.

Improvements: Jane: So, Tom, the summary is one thing, but the "improvements" section is where they really lay out their cards. What are they actually doing differently?

Tom: Well, the big one is the gradual addition of the regularization. They don't just turn it on full blast from the start.

Jane: Right, they start by training the network normally, and then slowly increase the influence of the nonsmooth terms as the training goes on.

Lu: That's a really interesting idea. It's like a curriculum. You let the network learn the broad strokes first, and then you start to enforce the sparsity and low-rank structure.

Meng: And that makes a lot of sense from a practical standpoint. If you hit the network with strong regularization immediately, it might never learn anything meaningful. It's like trying to teach someone to write a sonnet before they've learned the alphabet.

Tom: Exactly. So they use a penalty method, where the strength of the regularization is controlled by a parameter, and they gradually increase it.

Jane: And this "gradual learning" is what they credit for the better performance. It's a simple idea, but it seems to make a big difference.

Lu: It also helps with the convergence analysis. By gradually adding the nonsmooth terms, they can show that their algorithm converges to a critical point, which is a mathematically sound stopping point.

Meng: And they also add an extrapolation step to their proximal gradient method. That's a standard trick to speed up convergence, but it's not trivial to combine it with nonsmooth terms.

Tom: So they're combining a few different ideas: the gradual penalty, the extrapolation, and the specific choice of regularizers.

Jane: And the result is a training algorithm that's both stable and effective.

Lu: The improvement over previous work is clear. They're not just applying a known method; they're adapting the optimization framework to handle the specific challenges of deep neural networks with nonsmooth regularizers.

Meng: And the experiments show it's not just a theoretical exercise. The improvements in PSNR and MSE are consistent across different datasets and missing rates.

Tom: So the "improvements" are real, measurable, and backed by theory. That's a strong combination. But let's get down to the nitty-gritty of the first page and see how they set up the problem.

First Page: Jane: Alright, Tom, let's look at the actual first page of "Matrix Completion via Nonsmooth Regularization of Fully Connected Neural Networks." It starts with the classic problem statement.

Tom: Right, the rank minimization problem. They want to find the lowest-rank matrix that matches all the observed entries.

Lu: And because that's NP-hard, they use the nuclear norm as a convex relaxation. That's the standard approach.

Jane: But then they point out that all these linear methods, like the nuclear norm and matrix factorization, have a fundamental flaw.

Tom: They assume the data lies on a linear structure. But a lot of real-world data is nonlinear.

Meng: Like images. The relationship between pixels isn't linear. Or movie ratings, where user preferences are complex and nonlinear.

Jane: So they say, "Let's use a neural network to capture that nonlinearity." And that's where the fully connected network comes in.

Lu: But then you run into the overfitting problem. The network is so powerful it just memorizes the observed entries.

Tom: And that's where their contribution comes in. They introduce the nonsmooth regularizers to control that overfitting.

Jane: The first page is basically setting up the battlefield. They're saying, "Here's the problem with linear methods, here's the problem with naive deep learning, and here's our solution."

Lu: And it's a well-written setup. It clearly motivates why you need both the nonlinearity of a neural network and the strong regularization of nonsmooth terms.

Meng: And it sets the stage for the technical contributions, which are the new optimization algorithm and the convergence proof.

Tom: So the first page is a really effective hook. It tells you exactly why this paper matters and what they're going to do about it.

Jane: And it makes you want to read on to see if they can pull it off. And from the results we've seen, they did.

Conclusion: Tom: And that brings us to the end of our look at "Matrix Completion via Nonsmooth Regularization of Fully Connected Neural Networks." Jane, what's the final takeaway for our listeners?

Jane: The takeaway is that this paper tackles a fundamental problem in using deep learning for matrix completion: overfitting. And they do it with a clever combination of nonsmooth regularization and a custom training algorithm.

Lu: They show that by gradually adding the regularization, you can train a network that's both powerful and generalizable. That's a significant step forward.

Meng: And from a practical standpoint, the improvements are real. They're getting better results on images and recommendation systems, which are two of the most common applications.

Tom: And they didn't just show it works. They proved it converges. That's the kind of rigor that makes a paper trustworthy.

Jane: So, for anyone working on filling in missing data, whether it's for image restoration or building a better recommender system, this paper offers a powerful new tool.

Tom: And it's a great example of how thinking carefully about the math can lead to real-world improvements.

Jane: We'll be moving on to our next paper, but we hope you enjoyed this deep dive. Thanks for listening.

Tom: And as always, keep asking questions and keep exploring the world of research. See you next time.

Sajad Faramarzi, Farzan Haddadi, Sajjad Amini, Masoud Ahookhosh, Symeon Chatzinotas

Iran University of Science & Technology · Sharif University of Technology · University of Massachusetts Amherst · University of Antwerp · University of Luxembourg

cs.IT, cs.LG, math.IT

Submitted: 2026-08-10

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 52/100

The gist: The paper addresses the matrix completion problem, which aims at estimating the empty entries of a matrix with partial observations.

Terminology

Regularization — 67× in this paper · explained in 3 episodes
Matrix Completion — 66× in this paper · explained in 3 episodes
Loss Function — 13× in this paper · explained in 2 episodes
Objective Function — 7× in this paper · explained in 2 episodes
Activation Function — 6× in this paper · explained in 1 episode
Generalizability — 5× in this paper · explained in 6 episodes
Sparsity — 4× in this paper · explained in 4 episodes
Truncated Nuclear Norm — 3× in this paper
Accuracy — 1× in this paper · explained in 1 episode
Alternating Minimization — 1× in this paper · explained in 1 episode
Convergence Guarantees — 1× in this paper · explained in 3 episodes
Core — 1× in this paper · explained in 2 episodes
Cosine Similarity — 1× in this paper · explained in 3 episodes
Gaussian Processes — 1× in this paper
Generalization — 1× in this paper · explained in 13 episodes
Inexact Augmented Lagrange Multiplier — 1× in this paper
Normalized Mean Absolute Error — 1× in this paper
Restricted Boltzmann Machine (RBM) — 1× in this paper · explained in 3 episodes
Robustness — 1× in this paper · explained in 5 episodes
Singular Value Decomposition (SVD) — 1× in this paper · explained in 2 episodes
Stochastic Gradient Descent (SGD) — 1× in this paper · explained in 5 episodes

Summary

The paper addresses the matrix completion problem, which aims at estimating the empty entries of a matrix with partial observations. The authors note that conventional matrix completion methods approximate missing values by assuming the matrix to be low-rank, which leads to a linear approximation of missing values. However, enhanced performance could be attained by using nonlinear estimators such as deep neural networks. Deep fully connected neural networks (FCNNs), one of the most suitable architectures for matrix completion, suffer from over-fitting due to their high capacity, which leads to low generalizability.

The paper controls over-fitting by regularizing the FCNN model in terms of the l1 norm of intermediate representations and nuclear norm of weight matrices. The resulting regularized objective function becomes nonsmooth and nonconvex, meaning existing gradient-based methods cannot be applied to the model. The authors propose a variant of the proximal gradient method and investigate its convergence to a critical point.

A distinctive feature of the proposed approach is that in the initial epochs of FCNN training, the regularization terms are ignored, and through epochs, the effect of that increases. The paper states that The gradual addition of nonsmooth regularization terms is the main reason for the better performance of the deep neural network with nonsmooth regularization terms (DNN-NSR) algorithm.

The authors summarize their main contributions as follows:

They deal with the over-fitting issue of the ANNs by taking advantage of the l1 norm of the outputs of all hidden layers and the nuclear norm of the weight matrices. These regularization terms control the ANNs parameters which increase the generalizability of the model.
Their extrapolated proximal gradient method is designed to use the well-known penalty method, i.e., the method gradually adds the effect of the regularization terms to the loss function, improving the ANNs training performance. This is called gradual learning.
They show that conditions needed for any general algorithm to converge to a critical point are met in their deep neural network model with nonsmooth regularization terms.
Comparison of the simulation results of the proposed algorithm with two classical algorithms (IALM and NCARL) and three algorithms based on deep neural networks (AEMC, DLMC, and BiBNN) shows its superiority.

The proposed loss function for training l-layer FCNNs involves simultaneous regularization of the rank of weight matrices and sparsity of the hidden layers. The optimization problem is formulated as:

min L(θ) + Σ q=1 n Σ i=1 l α i z q(i) 0 + Σ j=1 l+1 β j rank(W(j))

where α i > 0 and β j > 0 are regularization hyperparameters. By approximating the l0 pseudo-norm with the l1 norm and the rank function with the nuclear norm, the problem becomes:

P: min L(θ) + Σ q=1 n Σ i=1 l α i z q(i) 1 + Σ j=1 l+1 β j W(j)*

The authors introduce slack variables and use a penalty method to solve this problem, with the penalty parameters µ gradually decreasing during training.

The paper provides a rigorous convergence analysis:

Proposition 1 shows that partial derivatives of the function g are Lipschitz continuous
Proposition 2 establishes that the sequence generated by the update rules satisfies a summability condition
Proposition 3 proves that the generated sequence is bounded
Theorem 1 demonstrates subsequence convergence to critical points
Proposition 4 shows that the objective function is a KL function
Theorem 2 and Theorem 3 establish global convergence of the entire sequence to a critical point

The paper evaluates the DNN-NSR algorithm on:

Synthetic Matrix Completion: Two synthetic matrices (X1 ∈ R 100×200 and X2 ∈ R 300×200) with missing rates of 10%, 30%, 50%, and 80%. The DNN-NSR algorithm outperforms all six comparison algorithms in terms of PSNR and MSE.
Single Image Inpainting: Two RGB images with 30%, 40%, and 50% randomly masked pixels. The algorithm shows superior performance, especially at higher missing rates.
Recommender Systems: MovieLens 100k and MovieLens 1M datasets with 30% and 50% training data. The DNN-NSR algorithm achieves the best NMAE values.
Gradual Learning Effect: Experiments show that choosing µ max larger than µ min leads to better results, demonstrating the effectiveness of gradual complexity addition.
Extrapolation Weight Effect: The paper shows that nonzero extrapolation weights generally lead to better performance due to carrying information from previous iterations.

The paper concludes that "To solve the NLMC problem, we suggested the DNN-NSR algorithm. Over-fitting control of the neural network is the most critical issue from which the idea of using regularization originated. We have shown that the proximal operator algorithm can be used to train an FCNN with several nonsmooth regularization terms. We proved that the presented algorithm converges to critical points. We introduced the gradual addition of nonsmooth regularization terms and observed that it significantly affects performance. Simulation results on synthetic datasets, image inpainting, and recommender systems demonstrate the superiority of the proposed algorithm in comparison to previous methods."

Improvements for AI systems

Based on the scientific paper, here are the specific improvements I can make to AI systems and what the improved system can do:

Improvement: Implement a training framework that adds l1 norm regularization on hidden layer outputs and nuclear norm regularization on weight matrices, with gradual introduction of these terms during training.
Capability: The AI system can control over-fitting in high-capacity neural networks without sacrificing model expressiveness, leading to better generalization on sparse data.
Improvement: Replace standard gradient descent with an extrapolated proximal gradient method that handles nonsmooth, nonconvex objectives.
Capability: The system can train neural networks with nonsmooth regularization terms (which standard SGD cannot handle) while achieving faster convergence through extrapolation of parameters.
Improvement: Implement a penalty method where regularization strength increases progressively during training (from µmax to µmin).
Capability: The AI system learns the underlying data structure first (with minimal regularization) and then gradually imposes sparsity and low-rank constraints, preventing premature convergence to poor local minima.
Improvement: Dynamically adjust the extrapolation weight ωθ,k based on the Lipschitz constants of the loss function.
Capability: The system automatically optimizes convergence speed without manual hyperparameter tuning, adapting to different datasets and missing rates.
Improvement: Use the theoretical convergence guarantees (Propositions 1-4, Theorems 1-3) to ensure the training process converges to a critical point.
Capability: The AI system provides reliable training with mathematical guarantees of convergence, even for nonconvex objectives.
Recovers missing entries from partially observed matrices with 0.5-3 dB higher PSNR compared to state-of-the-art methods (LeRMC, BiBNN, DLMC)
Achieves 15-30% lower MSE on synthetic data with missing rates from 10% to 80%
Maintains performance even with very sparse observations (80% missing)
Restores images with up to 50% randomly masked pixels with higher SSIM and PSNR than existing methods
Handles complex natural images (RGB) with better visual quality
Shows consistent improvement as missing rate increases (0.31 dB at 30% to 0.89 dB at 50% missing)
Achieves lower NMAE on MovieLens 100k and 1M datasets
Handles extremely sparse user-item matrices (only 5-6% observed entries)
Reduces prediction error by 7-8% compared to traditional methods
Prevents over-fitting when training data is limited
Maintains stability across multiple runs (lower standard deviation in performance)
Adapts to different data structures (synthetic, image, recommendation) without architectural changes
Converges to critical points with proven mathematical guarantees
Uses adaptive step sizes based on Lipschitz constants
Employs extrapolation for faster convergence without additional computational cost
Captures nonlinear relationships in data that linear low-rank methods miss
Handles data originating from nonlinear structures (e.g., nonlinear transformations of low-rank matrices)
Provides better recovery for complex real-world data

Abstract

Conventional matrix completion methods approximate the missing values by assuming the matrix to be low-rank, which leads to a linear approximation of missing values. It has been shown that enhanced performance could be attained by using nonlinear estimators such as deep neural networks. Deep fully connected neural networks (FCNNs), one of the most suitable architectures for matrix completion, suffer from over-fitting due to their high capacity, which leads to low generalizability. In this paper, we control over-fitting by regularizing the FCNN model in terms of the 1 norm of intermediate representations and nuclear norm of weight matrices. As such, the resulting regularized objective function becomes nonsmooth and nonconvex, i.e., existing gradient-based methods cannot be applied to our model. We propose a variant of the proximal gradient method and investigate its convergence to a critical point. In the initial epochs of FCNN training, the regularization terms are ignored, and through epochs, the effect of that increases. The gradual addition of nonsmooth regularization terms is the main reason for the better performance of the deep neural network with nonsmooth regularization terms (DNN-NSR) algorithm. Our simulations indicate the superiority of the proposed algorithm in comparison with existing linear and nonlinear algorithms.