  The model of low-dimensional manifold and sparse representation are two
well-known concise models that suggest each data can be described by a few
characteristics. Manifold learning is usually investigated for dimension
reduction by preserving some expected local geometric structures from the
original space to a low-dimensional one. The structures are generally
determined by using pairwise distance, e.g., Euclidean distance. Alternatively,
sparse representation denotes a data point as a linear combination of the
points from the same subspace. In practical applications, however, the nearby
points in terms of pairwise distance may not belong to the same subspace, and
vice versa. Consequently, it is interesting and important to explore how to get
a better representation by integrating these two models together. To this end,
this paper proposes a novel coding algorithm, called Locality-Constrained
Collaborative Representation (LCCR), which improves the robustness and
discrimination of data representation by introducing a kind of local
consistency. The locality term derives from a biologic observation that the
similar inputs have similar code. The objective function of LCCR has an
analytical solution, and it does not involve local minima. The empirical
studies based on four public facial databases, ORL, AR, Extended Yale B, and
Multiple PIE, show that LCCR is promising in recognizing human faces from
frontal views with varying expression and illumination, as well as various
corruptions and occlusions.
