This paper investigates the performance of a new multivariate method for tensor-based morphometry (TBM). Statistics on Riemannian manifolds are developed that exploit the full information in deformation tensor fields. In TBM, multiple brain images are warped to a common neuroanatomical template via 3-D nonlinear registration; the resulting deformation fields are analyzed statistically to identify group differences in anatomy. Rather than study the Jacobian determinant (volume expansion factor) of these deformations, as is common, we retain the full deformation tensors and apply a manifold version of Hotelling's $T(2) test to them, in a Log-Euclidean domain. In 2-D and 3-D magnetic resonance imaging (MRI) data from 26 HIV/AIDS patients and 14 matched healthy subjects, we compared multivariate tensor analysis versus univariate tests of simpler tensor-derived indices: the Jacobian determinant, the trace, geodesic anisotropy, and eigenvalues of the deformation tensor, and the angle of rotation of its eigenvectors. We detected consistent, but more extensive patterns of structural abnormalities, with multivariate tests on the full tensor manifold. Their improved power was established by analyzing cumulative p-value plots using false discovery rate (FDR) methods, appropriately controlling for false positives. This increased detection sensitivity may empower drug trials and large-scale studies of disease that use tensor-based morphometry.

We study the learnability of read-k-satisfy-j (RkSj) DNF formulas. These are boolean formulas in disjunctive normal form (DNF), in which the maximum number of occurrences of a variable is bounded by k, and the number of terms satisfied by any assignment is at most j. After motivating the investigation of this class of DNF formulas, we present an algorithm that for any unknown RkSj DNF formula to be learned, with high probability finds a logically equivalent DNF formula using the well-studied protocol of equivalence and membership queries. The algorithm runs in polynomial time for $k\cdot j=O({\log n\over\log\log n})$, where n is the number of input variables.

A polynomial-time algorithm is presented for exactly learning the class of read-k disjoint DNF formulas—boolean formulas in disjunctive normal form where each variable appears at most k) and every assignment to the variables satisfies at most one term of F. The (standard) protocol used allows the learning algorithm to query whether a given assignment of boolean variables satisfies the DNF formula to be learned (membership queries), as well as to obtain counterexamples to the correctness of its current hypothesis which can be any arbitrary DNF formula (equivalence queries). The formula output by the learning algorithm is logically equivalent to the formula to be learned. We show that this result also applies for a generalization of read-k disjoint DNF which we call read-k sat-j DNF; these are DNF formulas in which every variable appears at most k times and every assignment satisfies at most j terms.

A general technique is developed to obtain nonlearnability results in the model of exact learning from equivalence and membership queries. The technique is applied to show that, assuming NP not=co-NP, there does not exist a polynomial-time membership and equivalence query algorithm for exactly learning read-thrice DNF formulas-boolean formulas in disjunctive normal form where each variable appears at most three times. This result adds evidence to the conjecture that DNF is hard to learn in the membership and equivalence query model.>

A polynomial-time algorithm is presented for exactly learning the class of read-twice DNF formulas, i.e. Boolean formulas in disjunctive normal form where each variable appears at most twice. The (standard) protocol used allows the learning algorithm to query whether a given assignment of Boolean variables satisfies the DNF formula to be learned (membership queries), as well as to obtain counterexamples to the correctness of its current hypothesis which can be any arbitrary DNF formula (equivalence queries). The formula output by the learning algorithm is logically equivalent to the formula to be learned.>