We compute the first three coefficients of the asymptotic expansion of Zelditch. We also prove that in general, the k th coefficient is a polynomial of the curvature and its derivative of weight k .

Random backpropagation (RBP) is a variant of the backpropagation algorithm for training neural networks, where the transpose of the forward matrices are replaced by fixed random matrices in the calculation of the weight updates. It is remarkable both because of its effectiveness, in spite of using random matrices to communicate error information, and because it completely removes the taxing requirement of maintaining symmetric weights in a physical neural system. To better understand random backpropagation, we first connect it to the notions of local learning and learning channels. Through this connection, we derive several alternatives to RBP, including skipped RBP (SRPB), adaptive RBP (ARBP), sparse RBP, and their combinations (e.g. ASRBP) and analyze their computational complexity. We then study their behavior through simulations using the MNIST and CIFAR-10 bechnmark datasets. These simulations show that most of these variants work robustly, almost as well as backpropagation, and that multiplication by the derivatives of the activation functions is important. As a follow-up, we study also the low-end of the number of bits required to communicate error information over the learning channel. We then provide partial intuitive explanations for some of the remarkable properties of RBP and its variations. Finally, we prove several mathematical results, including the convergence to fixed points of linear chains of arbitrary length, the convergence to fixed points of linear autoencoders with decorrelated data, the long-term existence of solutions for linear systems with a single hidden layer and convergence in special cases, and the convergence to fixed points of non-linear chains, when the derivative of the activation functions is included.

After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corollary, we prove the formula for the BCOV invariant of quintic mirror threefolds conjectured by Bershadsky-Cecotti-Ooguri-Vafa. Contents 1. Introduction 2. Calabi-Yau varieties with at most one ordinary double point 3. Quillen metrics 4. The BCOV invariant of Calabi-Yau manifolds 5. The singularity of the Quillen metric on the BCOV bundle 6. The cotangent sheaf of the Kuranishi space 7. Behaviors of the Weil-Petersson metric and the Hodge metric 8. The singularity of the BCOV invariant I -the case of ODP 9. The singularity of the BCOV invariant II -general degenerations 10. The curvature current of the BCOV invariant 11. The BCOV invariant of Calabi-Yau threefolds with h 1,2 = 1 12. The BCOV invariant of quintic mirror threefolds 13. The BCOV invariant of FHSV threefolds 1 ANALYTIC TORSION FOR CALABI-YAU THREEFOLDS 5an arbitrary Calabi-Yau manifold of arbitrary dimension, which we obtain using determinants of cohomologies [28], Quillen metrics [11], [44], and a Bott-Chern class like A(·). Then the BCOV Hermitian line of a Calabi-Yau manifold depends only on the complex structure of the manifold. The Hodge diamond of Calabi-Yau threefolds are so simple that the BCOV Hermitian line reduces to the scalar invariant τ BCOV in the case of threefolds. Hence Eq. (1.1) on P 1 \ D is deduced from the curvature formula for the BCOV Hermitian line bundles. (See Sect. 4).(b) To establish the formula for log τ BCOV near D, we fix a specific holomorphic extension of the BCOV bundle from P 1 \D to P 1 , which we call the Kähler extension. (See Sect. 5.) Since τ BCOV is the ratio of the Quillen metric and the L 2 -metric on the BCOV bundle, it suffices to determine the singularities of the Quillen metric and the L 2 -metric on the extended BCOV bundle. We determine the singularity of the Quillen metric on the extended BCOV bundle with respect to the metric on T X /P 1 induced from a Kähler metric on X . The anomaly formula for Quillen metrics of Bismut-Gillet-Soulé [11] and a formula for the singularity of Quillen metrics [9], [61] play the central role. (See Sect. 5.).(c) By the smoothness of Def(X ψ ) at ψ ∈ D * [26], [45], [54], the behavior of the L 2 -metric on the extended BCOV bundle near D * is determined by the singularity of Ω WP near D * , which was computed by Tian [54]. (See Sects. 6,7,8.) To determine the behavior of the L 2 metric on the extended BCOV bundle at ψ = ∞, one may assume that π : X → P 1 is semi-stable at ψ = ∞ by Mumford [27]. We consider another holomorphic extension of the BCOV bundle, i.e., the canonical extension in Hodge theory [48]. With respect to the canonical extension, the L 2 -metric has at most an algebraic singularity at ψ = ∞ by Schmid [48]. Comparing the two extensions, we sh...

In this paper, we proved the Normal Scalar Curvature Conjecture and the Böttcher-Wenzel Conjecture. We developed a new Bochner formula and it becomes useful with the first conjecture we proved. Using the results, we established some new pinching theorems for minimal submanifolds in spheres. Published by Elsevier Inc.

We study the local properties of the moduli space of a polarized Calabi-Yau manifold. Let U be a neighborhood of the moduli space. Then we know the universal covering space V of U is a smooth manifold. Suppose D is the classifying space of a polarized Calabi-Yau manifold with the automorphism group G . Then we prove that the map from V to D induced by the period map is a pluriharmonic map. We also give a Kähler metric on V , which is called the Hodge metric. We prove that the Ricci curvature of the Hodge metric is negative away from zero. We also proved the nonexistence of the Kähler metric on the classifying space of a Calabi-Yau threefold which is invariant under a cocompact lattice of G .