Monday, September 03, 2012

Implementation: Bilinear modelling via Augmented Lagrange Multipliers (BALM)

After the pretty long list of implementations made available over this past yearAlessio Del Bue walks in that same distinguished path of reproducible research: :

Dear Igor,

again congratulations for all your efforts in putting order in all these factorizations. We have a contribution and I would ask if it could be added to your list. It is called BALM and it is on this month PAMI issue:

A. Del Bue, J. Xavier, L. Agapito, and M. Paladini, "Bilinear Modeling via Augmented Lagrange Multipliers (BALM)," Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol. 34, iss. 8, pp. 1496-1508, 2012.

It solves this matrix completion problem:

A = H.*DX
subject to D \in \mathcal{M}

D unknown
X unknown
H is a known mask matrix
D belongs to a matrix manifold \mathcal{M}

and it is locally convergent for a wide category of manifolds. If you need more details or you can drop me an email!

Thank you,

Thanks Alessio !

Abstract—This paper presents a unified approach to solve different bilinear factorization problems in computer vision in the presence of missing data in the measurements. The problem is formulated as a constrained optimization where one of the factors must lie on a specific manifold. To achieve this, we introduce an equivalent reformulation of the bilinear factorization problem that decouples the core bilinear aspect from the manifold specificity. We then tackle the resulting constrained optimization problem via Augmented Lagrange Multipliers. The strength and the novelty of our approach is that this framework can handle seamlessly different computer vision problems. The algorithm is such that only a projector onto the manifold constraint is needed. We present experiments and results for some popular factorization problems in computer vision such as rigid, non-rigid and articulated Structure from Motion; photometric stereo and 2D-3D non-rigid registration.
I should add it shortly to the Matrix Factorization Jungle page.

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