127 research outputs found
Stability of matrix factorization for collaborative filtering
We study the stability vis a vis adversarial noise of matrix factorization
algorithm for matrix completion. In particular, our results include: (I) we
bound the gap between the solution matrix of the factorization method and the
ground truth in terms of root mean square error; (II) we treat the matrix
factorization as a subspace fitting problem and analyze the difference between
the solution subspace and the ground truth; (III) we analyze the prediction
error of individual users based on the subspace stability. We apply these
results to the problem of collaborative filtering under manipulator attack,
which leads to useful insights and guidelines for collaborative filtering
system design.Comment: ICML201
Weighted Low Rank Approximation for Background Estimation Problems
Classical principal component analysis (PCA) is not robust to the presence of
sparse outliers in the data. The use of the norm in the Robust PCA
(RPCA) method successfully eliminates the weakness of PCA in separating the
sparse outliers. In this paper, by sticking a simple weight to the Frobenius
norm, we propose a weighted low rank (WLR) method to avoid the often
computationally expensive algorithms relying on the norm. As a proof
of concept, a background estimation model has been presented and compared with
two norm minimization algorithms. We illustrate that as long as a
simple weight matrix is inferred from the data, one can use the weighted
Frobenius norm and achieve the same or better performance
Projective Bundle Adjustment from Arbitrary Initialization Using the Variable Projection Method
Bundle adjustment is used in structure-from-motion pipelines as final refinement stage requiring a sufficiently good initialization to reach a useful local mininum. Starting from an arbitrary initialization almost always gets trapped in a poor minimum. In this work we aim to obtain an initialization-free approach which returns global minima from a large proportion of purely random starting points. Our key inspiration lies in the success of the Variable Projection (VarPro) method for affine factorization problems, which have close to 100% chance of reaching a global minimum from random initialization. We find empirically that this desirable behaviour does not directly carry over to the projective case, and we consequently design and evaluate strategies to overcome this limitation. Also, by unifying the affine and the projective camera settings, we obtain numerically better conditioned reformulations of original bundle adjustment algorithms
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