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**1 - 4**of**4**### Reduced row echelon form and non-linear approximation for subspace segmentation and high-dimensional data clustering

, 2014

"... Given a set of data W = {w 1 , . . . , w N } ∈ R D drawn from a union of subspaces, we focus on determining a nonlinear model of the form U = i∈I S i , where {S i ⊂ R D } i∈I is a set of subspaces, that is nearest to W. The model is then used to classify W into clusters. Our approach is based on th ..."

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Given a set of data W = {w 1 , . . . , w N } ∈ R D drawn from a union of subspaces, we focus on determining a nonlinear model of the form U = i∈I S i , where {S i ⊂ R D } i∈I is a set of subspaces, that is nearest to W. The model is then used to classify W into clusters. Our approach is based on the binary reduced row echelon form of data matrix, combined with an iterative scheme based on a non-linear approximation method. We prove that, in absence of noise, our approach can find the number of subspaces, their dimensions, and an orthonormal basis for each subspace S i . We provide a comprehensive analysis of our theory and determine its limitations and strengths in presence of outliers and noise.

### Topk Preferences in High Dimensions

"... In many applications, users are interested only in a small number (say,k) of “top ” objects from a large set. For each user, objects are ranked based on his or her individual preference. If the objects have multiple numeric attributes, a user preference is often specified ..."

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In many applications, users are interested only in a small number (say,k) of “top ” objects from a large set. For each user, objects are ranked based on his or her individual preference. If the objects have multiple numeric attributes, a user preference is often specified

### Subspace Clustering

"... Abstract We present a new approach to rigid-body mo-tion segmentation from two views. We use a previously de-veloped nonlinear embedding of two-view point correspon-dences into a 9-dimensional space and identify the differ-ent motions by segmenting lower-dimensional subspaces. In order to overcome n ..."

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Abstract We present a new approach to rigid-body mo-tion segmentation from two views. We use a previously de-veloped nonlinear embedding of two-view point correspon-dences into a 9-dimensional space and identify the differ-ent motions by segmenting lower-dimensional subspaces. In order to overcome nonuniform distributions along the subspaces, whose dimensions are unknown, we suggest the novel concept of global dimension and its minimization for clustering subspaces with some theoretical motivation. We propose a fast projected gradient algorithm for minimiz-ing global dimension and thus segmenting motions from 2-views. We develop an outlier detection framework around the proposed method, and we present state-of-the-art results on outlier-free and outlier-corrupted two-view data for seg-menting motion.