Locality preserving projections

He, X., Niyogi, P. Locality Preserving Projections. In Advances in Neural Information Processing Systems 16 Proceedings of the 2003 Conference (NIPS), pp. 153-160. MIT Press... 

One further point of interest with regards to Laplacian Eigenmaps is the algorithm s linearised variant. Locality Preserving Projections (LPP) [15]. LPP seeks to compute a transformation matrix using the graph Laplacian that maps the data points into a subspace. Specifically, LPP seeks to minimise... 

Proof. Note that Lf is computed using a flat resolution, and a flat object is preserved by /. If is a flat morphism of schemes, then g is exact. Thus using a spectral sequence argument, it is easy to see that the question is local both on Y and X. So we may assume that X = SpecB and Y = Spec A are affine. If r(y, Xf) = M and F —> M is an A-projective resolution, then... 

There are other problems too. Some of the projects closed down quite quickly without much time to consider the problem of knowledge preservation. So the information may exist only as paper documents in the local language. Where knowledge preservation has been undertaken, it has been done so against the background of a project closure, with the belief that this information would not be required for many years to come. So little consideration has been given to accessibility of information to ongoing projects. 

It is desirable to construct formulations and numerical methods which exactly (i.e. up to rounding error) preserve the total momentum from step to step. One obvious approaeh to this problem is to simply project the momenta onto the linear momenrnm constraint at the end of each step (or after some number of steps). Such a projection introduces potential issues in terms of convergence order and would certainly complicate the analyses presented thus far in this book. Moreover, the optimal choice of projection is unclear and it is easy to define poor schemes (for example, modifying always the momentum of just the first particle in order to balance all the remaining components) which are likely to introduce artifacts (bias) in simulation. For this reason, there is interest in building in momentum conservation into the equations of motion (and indeed the integrator). Ideally this should be done in a localized and homogeneous way so that momentum is not transferred by a nonphysical mechanism between distant particles. 

Commercial exploitation of natural products from the tropical forests may, in fact, be the only way to preserve this habitat and avoid its destruction by local inhabitants in search of conditions for survival or by commercial enterprises after a short to medium-term gain. A number of projects now under way indicate that rational exploitation is possible without permanent damage to the environment and scientific management of degraded areas may accelerate their recovery. 

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