Non-negative Matrix Factorization via Archetypal Analysis
نویسندگان
چکیده
Given a collection of data points, non-negative matrix factorization (NMF) suggests to express them as convex combinations of a small set of ‘archetypes’ with non-negative entries. This decomposition is unique only if the true archetypes are non-negative and sufficiently sparse (or the weights are sufficiently sparse), a regime that is captured by the separability condition and its generalizations. In this paper, we study an approach to NMF that can be traced back to the work of Cutler and Breiman [CB94] and does not require the data to be separable, while providing a generally unique decomposition. We optimize the trade-off between two objectives: we minimize the distance of the data points from the convex envelope of the archetypes (which can be interpreted as an empirical risk), while minimizing the distance of the archetypes from the convex envelope of the data (which can be interpreted as a data-dependent regularization). The archetypal analysis method of [CB94] is recovered as the limiting case in which the last term is given infinite weight. We introduce a ‘uniqueness condition’ on the data which is necessary for exactly recovering the archetypes from noiseless data. We prove that, under uniqueness (plus additional regularity conditions on the geometry of the archetypes), our estimator is robust. While our approach requires solving a non-convex optimization problem, we find that standard optimization methods succeed in finding good solutions both for real and synthetic data.
منابع مشابه
Approximate Nonnegative Matrix Factorization via Alternating Minimization
In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix V ∈ R + find, for assigned k, nonnegative matrices W ∈ R + and H ∈ R k×n + such that V = WH . Exact, non trivial, nonnegative factorizations do not always exist, hence it is interesting to pose the approximate NMF problem. The criterion which is commonly employed is I-divergen...
متن کاملIterative Weighted Non-smooth Non-negative Matrix Factorization for Face Recognition
Non-negative Matrix Factorization (NMF) is a part-based image representation method. It comes from the intuitive idea that entire face image can be constructed by combining several parts. In this paper, we propose a framework for face recognition by finding localized, part-based representations, denoted “Iterative weighted non-smooth non-negative matrix factorization” (IWNS-NMF). A new cost fun...
متن کاملMatrix factorization methods: application to Thermal NDT/E
A typical problem in Thermal Nondestructive Testing/Evaluation (TNDT/E) is that of unsupervised feature extraction from the experimental data. Matrix factorization methods (MFMs) are mathematical techniques well suited for this task. In this paper we present the application of three MFMs: Principal Component Analysis (PCA), Non-negative Matrix Factorization (NMF), and Archetypal Analysis (AA). ...
متن کاملHierarchical Convex NMF for Clustering Massive Data
We present an extension of convex-hull non-negative matrix factorization (CH-NMF) which was recently proposed as a large scale variant of convex non-negative matrix factorization or Archetypal Analysis. CH-NMF factorizes a non-negative data matrix V into two nonnegative matrix factors V ≈ WH such that the columns of W are convex combinations of certain data points so that they are readily inter...
متن کاملNonnegative Matrix Factorization and I-Divergence Alternating Minimization
In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix V ∈ R + find, for assigned k, nonnegative matrices W ∈ R + and H ∈ R k×n + such that V = WH . Exact, non trivial, nonnegative factorizations do not always exist, hence it is interesting to pose the approximate NMF problem. The criterion which is commonly employed is I-divergen...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1705.02994 شماره
صفحات -
تاریخ انتشار 2017