Notes on PCA, Regularization, Sparsity and Support Vector Machines

We derive a new representation for a function as a linear combination of local correlation kernels at optimal sparse locations and discuss its relation to PCA, regularization, sparsity principles and Support Vector Machines. We first review previous results for the approximation of a function from discrete data (Girosi, 1998) in the context of Vapnik"s feature space and dual representation (Vapnik, 1995). We apply them to show 1) that a standard regularization functional with a stabilizer defined in terms of the correlation function induces a regression function in the span of the feature space of classical Principal Components and 2) that there exist a dual representations of the regression function in terms of a regularization network with a kernel equal to a generalized correlation function. We then describe the main observation of the paper: the dual representation in terms of the correlation function can be sparsified using the Support Vector Machines (Vapnik, 1982) technique and this operation is equivalent to sparsify a large dictionary of basis functions adapted to the task, using a variation of Basis Pursuit De-Noising (Chen, Donoho and Saunders, 1995; see also related work by Donahue and Geiger, 1994; Olshausen and Field, 1995; Lewicki and Sejnowski, 1998). In addition to extending the close relations between regularization, Support Vector Machines and sparsity, our work also illuminates and formalizes the LFA concept of Penev and Atick (1996). We discuss the relation between our results, which are about regression, and the different problem of pattern classification.

Core-Shell Assisted Bimetallic Assembly of Pt and Ru Nanoparticles by DNA Hybridization

We have discovered that the current protocols to assemble Au nanoparticles based on DNA hybridization do not work well with the small metal nanoparticles (e.g. 5 nm Au, 3.6 nm Pt and 3.2 nm Ru particles). Further investigations revealed the presence of strong interaction between the oligonucleotide backbone and the surface of the small metal nanoparticles. The oligonucleotides in this case are recumbent on the particle surface and are therefore not optimally oriented for hybridization. The nonspecific adsorption of oligonucleotides on small metal nanoparticles must be overcome before DNA hybridization can be accepted as a general assembly method. Two methods have been suggested as possible solutions to this problem. One is based on the use of stabilizer molecules which compete with the oligonucleotides for adsorption on the metal nanoparticle surface. Unfortunately, the reported success of this approach in small Au nanoparticles (using K₂BSPP) and Au films (using 6-mercapto-1-hexanol) could not be extended to the assembly of Pt and Ru nanoparticles by DNA hybridization. The second approach is to simply use larger metal particles. Indeed most reports on the DNA hybridization induced assembly of Au nanoparticles have made use of relatively large particles (>10 nm), hinting at a weaker non-specific interaction between the oligonucleotides and large Au nanoparticles. However, most current methods of nanoparticle synthesis are optimized to produce metal nanoparticles only within a narrow size range. We find that core-shell nanoparticles formed by the seeded growth method may be used to artificially enlarge the size of the metal particles to reduce the nonspecific binding of oligonucleotides. We demonstrate herein a core-shell assisted growth method to assemble Pt and Ru nanoparticles by DNA hybridization. This method involves firstly synthesizing approximately 16 nm core-shell Ag-Pt and 21 nm core-shell Au-Ru nanoparticles from 9.6 nm Ag seeds and 17.2 nm Au seeds respectively by the seed-mediated growth method. The core-shell nanoparticles were then functionalized by complementary thiolated oligonucleotides followed by aging in 0.2 M PBS buffer for 6 hours. The DNA hybridization induced bimetallic assembly of Pt and Ru nanoparticles could then be carried out in 0.3 M PBS buffer for 10 hours.