A Unified Convergence Analysis of the Multiplicative Update Algorithm for Regularized NMF with General Divergences

نویسندگان

  • Renbo Zhao
  • Vincent Y. F. Tan
چکیده

The multiplicative update (MU) algorithm has been used extensively to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizations. However, theoretical convergence guarantees have only been derived for a few special divergences and without regularizers. We provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and with l1 and Tikhonov regularizations. Our result shows the sequence of iterates (i.e., pairs of basis and coefficient matrices) produced by the MU algorithm converges to the set of stationary points of the NMF (optimization) problem. Our proof strategy has the potential to open up new avenues for analyzing similar problems.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Efficient Multiplicative Updates for Support Vector Machines

The dual formulation of the support vector machine (SVM) objective function is an instance of a nonnegative quadratic programming problem. We reformulate the SVM objective function as a matrix factorization problem which establishes a connection with the regularized nonnegative matrix factorization (NMF) problem. This allows us to derive a novel multiplicative algorithm for solving hard and sof...

متن کامل

Multiplicative Updates for Elastic Net Regularized Convolutional NMF Under $\beta$-Divergence

We generalize the convolutional NMF by taking the β-divergence as the loss function, add a regularizer for sparsity in the form of an elastic net, and provide multiplicative update rules for its factors in closed form. The new update rules embed the β-NMF, the standard convolutional NMF, and sparse coding alias basis pursuit. We demonstrate that the originally published update rules for the con...

متن کامل

Stability analysis of multiplicative update algorithms and application to non-negative matrix factorization Analyse de la stabilité des règles de mises à jour multiplicatives et application à la factorisation en matrices positives

Multiplicative update algorithms have encountered a great success to solve optimization problems with nonnegativity constraints, such as the famous non-negative matrix factorization (NMF) and its many variants. However, despite several years of research on the topic, the understanding of their convergence properties is still to be improved. In this paper, we show that Lyapunov’s stability theor...

متن کامل

A Converged Algorithm for Tikhonov Regularized Nonnegative Matrix Factorization with Automatic Regularization Parameters Determination

We present a converged algorithm for Tikhonov regularized nonnegative matrix factorization (NMF). We specially choose this regularization because it is known that Tikhonov regularized least square (LS) is the more preferable form in solving linear inverse problems than the conventional LS. Because an NMF problem can be decomposed into LS subproblems, it can be expected that Tikhonov regularized...

متن کامل

Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative Matrix Factorization

We propose a class of multiplicative algorithms for Nonnegative Matrix Factorization (NMF) which are robust with respect to noise and outliers. To achieve this, we formulate a new family generalized divergences referred to as the Alpha-Beta-divergences (AB-divergences), which are parameterized by the two tuning parameters, alpha and beta, and smoothly connect the fundamental Alpha-, Betaand Gam...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2016