Geometry of good sets in n-fold Cartesian product

We propose here a multidimensional generalisation of the notion of link introduced in our previous papers and we discuss some consequences for simplicial measures and sums of function algebras.Comment: 17 pages, no figures, no table

Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method

We give a new approach to the dictionary learning (also known as "sparse coding") problem of recovering an unknown $n\times m$ matrix $A$ (for $m \geq n$) from examples of the form $$y = Ax + e,$$ where $x$ is a random vector in $\mathbb R^m$ with at most $\tau m$ nonzero coordinates, and $e$ is a random noise vector in $\mathbb R^n$ with bounded magnitude. For the case $m=O(n)$, our algorithm recovers every column of $A$ within arbitrarily good constant accuracy in time $m^{O(\log m/\log(\tau^{-1}))}$, in particular achieving polynomial time if $\tau = m^{-\delta}$ for any $\delta>0$, and time $m^{O(\log m)}$ if $\tau$ is (a sufficiently small) constant. Prior algorithms with comparable assumptions on the distribution required the vector $x$ to be much sparser---at most $\sqrt{n}$ nonzero coordinates---and there were intrinsic barriers preventing these algorithms from applying for denser $x$. We achieve this by designing an algorithm for noisy tensor decomposition that can recover, under quite general conditions, an approximate rank-one decomposition of a tensor $T$, given access to a tensor $T'$ that is $\tau$-close to $T$ in the spectral norm (when considered as a matrix). To our knowledge, this is the first algorithm for tensor decomposition that works in the constant spectral-norm noise regime, where there is no guarantee that the local optima of $T$ and $T'$ have similar structures. Our algorithm is based on a novel approach to using and analyzing the Sum of Squares semidefinite programming hierarchy (Parrilo 2000, Lasserre 2001), and it can be viewed as an indication of the utility of this very general and powerful tool for unsupervised learning problems

Nearly Optimal Private Convolution

We study computing the convolution of a private input $x$ with a public input $h$, while satisfying the guarantees of $(\epsilon, \delta)$-differential privacy. Convolution is a fundamental operation, intimately related to Fourier Transforms. In our setting, the private input may represent a time series of sensitive events or a histogram of a database of confidential personal information. Convolution then captures important primitives including linear filtering, which is an essential tool in time series analysis, and aggregation queries on projections of the data. We give a nearly optimal algorithm for computing convolutions while satisfying $(\epsilon, \delta)$-differential privacy. Surprisingly, we follow the simple strategy of adding independent Laplacian noise to each Fourier coefficient and bounding the privacy loss using the composition theorem of Dwork, Rothblum, and Vadhan. We derive a closed form expression for the optimal noise to add to each Fourier coefficient using convex programming duality. Our algorithm is very efficient -- it is essentially no more computationally expensive than a Fast Fourier Transform. To prove near optimality, we use the recent discrepancy lowerbounds of Muthukrishnan and Nikolov and derive a spectral lower bound using a characterization of discrepancy in terms of determinants

Rational Design of Peptide Ligands Based on Knob−Socket Protein Packing Model Using CD13 as a Prototype Receptor

Structure-based computational peptide design methods have gained significant interest in recent years owing to the availability of structural insights into protein–protein interactions obtained from the crystal structures. The majority of these approaches design new peptide ligands by connecting the crucial amino acid residues from the protein interface and are generally not based on any predicted receptor–ligand interaction. In this work, a peptide design method based on the Knob–Socket model was used to identify the specific ligand residues packed into the receptor interface. This method enables peptide ligands to be designed rationally by predicting amino acid residues that will fit best at the binding site of the receptor protein. In this, specific peptide ligands were designed for the model receptor CD13, overexpression of which has been observed in several cancer types. From the initial library of designed peptides, three potential candidates were selected based on simulated energies in the CD13 binding site using the programs molecular operating environment and AutoDock Vina. In the CD13 enzymatic activity inhibition assay, the three identified peptides exhibited 2.7–7.4 times lower IC50 values (GYPAY, 227 μM; GFPAY, 463 μM; GYPAVYLF, 170 μM) as compared to the known peptide ligand CNGRC (C1–C5) (1260 μM). The apparent binding affinities of the peptides (GYPAY, Ki = 54.0 μM; GFPAY, Ki = 74.3 μM; GYPAVYLF, Ki = 38.8 μM) were 10–20 times higher than that of CNGRC (C1–C5) (Ki = 773 μM). The double reciprocal plots from the steady-state enzyme kinetic assays confirmed the binding of the peptides to the intended active site of CD13. The cell binding and confocal microscopy assays showed that the designed peptides selectively bind to the CD13 on the cell surface. Our study demonstrates the feasibility of a Knob–Socket-based rational design of novel peptide ligands in improving the identification of specific binding versus current more labor-intensive methods