A D-optimal design is generated by an iterative search algorithmand seeks to minimize the covariance of the parameter estimates fora specified model. This is equivalent to maximizing the determinant D =XTX,where X is the design matrix of model terms (thecolumns) evaluated at specific treatments in the design space (therows). Unlike traditional designs, D-optimal designs do not requireorthogonal design matrices, and as a result, parameter estimates maybe correlated. Parameter estimates may also be locally, but not globally,D-optimal.
We compute the c-optimal design (c=[1,2,3,4,5])for the observation matrices A[i].T from the variable A defined above.The results below suggest that we should allocate 12.8% of theexperimental effort on design point #5, and 87.2% on the design point #7.
D-Optimal Matrices
When the observation matrices are row vectors (single-response framework),the SOCP above reduces to a simple LP, because the variables are scalar.We solve below the LP for the case where there are 11available design points, corresponding to the columns of the matricesA[4], A[5], A[6], and A[7][:,:-1] defined in the preambule.
We compute the A-optimal designfor the observation matrices A[i].T defined in the preambule.The optimal design allocates24.9% on design point #3,14.2% on point #4,8.51% on point #5,12.1% on point #6,13.2% on point #7,and 27.0% on point #8.
A-optimal designs can also be computed by SOCPwhen the vector of weights is subjectto several linear constraints.To give an example, we compute the A-optimal design forthe observation matrices given in the preambule, when the weightsmust satisfy: and .This problem has the following SOCP formulation:
Results: In this article, we present a rigorous approach to biclustering, OREO, which is based on the Optimal RE-Ordering of the rows and columns of a data matrix so as to globally minimize the dissimilarity metric. The physical permutations of the rows and columns of the data matrix can be modeled as either a network flow problem or a traveling salesman problem. Cluster boundaries in one dimension are used to partition and re-order the other dimensions of the corresponding submatrices to generate biclusters. The performance of OREO is tested on (a) metabolite concentration data, (b) an image reconstruction matrix, (c) synthetic data with implanted biclusters, and gene expression data for (d) colon cancer data, (e) breast cancer data, as well as (f) yeast segregant data to validate the ability of the proposed method and compare it to existing biclustering and clustering methods.
Conclusion: We demonstrate that this rigorous global optimization method for biclustering produces clusters with more insightful groupings of similar entities, such as genes or metabolites sharing common functions, than other clustering and biclustering algorithms and can reconstruct underlying fundamental patterns in the data for several distinct sets of data matrices arising in important biological applications.
A quantum version of the Monge-Kantorovich optimal transport problem is analyzed. The transport cost is minimized over the set of all bipartite coupling states ρAB such that both of its reduced density matrices ρA and ρB of dimension N are fixed. We show that, selecting the quantum cost matrix to be proportional to the projector on the antisymmetric subspace, the minimal transport cost leads to a semidistance between ρA and ρB, which is bounded from below by the rescaled Bures distance and from above by the root infidelity. In the single-qubit case, we provide a semianalytic expression for the optimal transport cost between any two states and prove that its square root satisfies the triangle inequality and yields an analog of the Wasserstein distance of the order of 2 on the set of density matrices. We introduce an associated measure of proximity of quantum states, called swap fidelity, and discuss its properties and applications in quantum machine learning.
2ff7e9595c
Comments