Approximate matrix factorization techniques with both nonnegativity and orthogonality constraints, referred to as orthogonal nonnegative matrix factorization (ONMF), have been recently introduced and shown to work remarkably well for clustering tasks such as document classification. In this paper, we introduce two new methods to solve ONMF. First, we show mathematical equivalence between ONMF and a weighted variant of spherical k-means, from which we derive our first method, a simple EM-like algorithm. This also allows us to determine when ONMF should be preferred to k-means and spherical k-means. Our second method is based on an augmented Lagrangian approach. Standard ONMF algorithms typically enforce nonnegativity for their iterates while trying to achieve orthogonality at the limit (e.g., using a proper penalization term or a suitably chosen search direction). Our method works the opposite way: orthogonality is strictly imposed at each step while nonnegativity is asymptotically obtained, using a quadratic penalty. Finally, we show that the two proposed approaches compare favorably with standard ONMF algorithms on synthetic, text and image data sets.

Brain single-photon-emission-computerized tomography (SPECT) with 123I-ioflupane (123I-FP-CIT) is useful to diagnose Parkinson disease (PD). To investigate the diagnostic performance of 123I-FP-CIT brain SPECT with semiquantitative analysis by Basal Ganglia V2 software (BasGan), we evaluated semiquantitative data of patients with suspect of PD by a support vector machine classifier (SVM), a powerful supervised classification algorithm.123I-FP-CIT SPECT with BasGan analysis was performed in 90 patients with suspect of PD showing mild symptoms (bradykinesia-rigidity and mild tremor). PD was confirmed in 56 patients, 34 resulted non-PD (essential tremor and drug-induced Parkinsonism). A clinical follow-up of at least 6 months confirmed diagnosis. To investigate BasGan diagnostic performance we trained SVM classification models featuring different descriptors using both a “leave-one-out” and a “five-fold” method. In the first study we used as class descriptors the semiquantitative radiopharmaceutical uptake values in the left (L) and right (R) putamen (P) and in the L and R caudate nucleus (C) for a total of 4 descriptors (CL, CR, PL, PR). In the second study each patient was described only by CL and CR, while in the third by PL and PR descriptors. Age was added as a further descriptor to evaluate its influence in the classification performance.123I-FP-CIT SPECT with BasGan analysis reached a classification performance higher than 73.9% in all the models. Considering the “Leave-one-out” method, PL and PR were better predictors (accuracy of 91% for all patients) than CL and CR descriptors; using PL, PR, CL, and CR diagnostic accuracy was similar to that of PL and PR descriptors in the different groups. Adding age as a further descriptor accuracy improved in all the models. The best results were obtained by using all the 5 descriptors both in PD and non-PD subjects (CR and CL + PR and PL + age = 96.4% and 94.1%, respectively). Similar results were observed for the “five-fold” method.123I-FP-CIT SPECT with BasGan analysis using SVM classifier was able to diagnose PD. Putamen was the most discriminative descriptor for PD and the patient age influenced the classification accuracy.