Robust Non-Negative Factorization using L11 norm and its application in image clustering Kai LIU Outline • • • • 1. 2. 3. 4. Introduction of NMF Why we need robust NMF My theoretical work Experiments and results Introduction • Non-negative Matrix Factorization has good interpretation: Why Robust NMF? • Traditional NMF: • Object : 2 min X FG F st.G T * G I • Using the least square error as the object, the function is unstable with noise, outliers. Outliers My method • Change the Objective to: min X FG 1,1 st.G T * G I • The reason is the object is the sum of absolute value of every entry in X-F*G, while traditional NMF aims to minimize the square Algorithm • Updating algorithm: • Where The idea • 1. First to prove X 1,1 trace( X * D * X T ) • 2. take derivative of the trace with respect to F and G(here X = X – F*G and X is constant) • 3. prove the convergence of the algorithm • 4. prove the correctness of the algorithm Experiments on ATNT • L2 algorithm: Experiments on ATNT • L1 algorithm: Experiments on Caltech 101 • We do the image clustering on Caltech 101, and here is a table(clustering accuracy) we get: L1 L2 kmeans Set 1 0.4833 0.4167 0.4333 Set 2 0.5167 0.4167 0.4667 Set 3 0.5167 0.3833 0.4000 Set 4 0.5000 0.4167 0.4500 Set 5 0.4500 0.3833 0.3833 Set 6 0.4833 0.4167 0.4000 Set 7 0.4833 0.4333 0.4333 Set 8 0.5833 0.4833 0.5167 Set 9 0.4333 0.3833 0.3833 Conclusion • By using L1 norm NMF, the feature(F) we get is better(more clear) than that of L2 norm • L1 NMF in many situations have better clustering accuracy(G) than L2 norm NMF and kmeans. Questions and Answers Thank you!