Higher spectral flow and an entire bivariant JLO cocycle

Given a smooth fibration of closed manifolds and a family of generalised Dirac operators along the fibers, we define an associated bivariant JLO cocycle. We then prove that, for any $\ell \geq 0$, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the $C^{\ell+1}$ topology and functions on the base manifold with the $C^\ell$ topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fr\'echet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow

Stochastic expansions using continuous dictionaries: L\'{e}vy adaptive regression kernels

This article describes a new class of prior distributions for nonparametric function estimation. The unknown function is modeled as a limit of weighted sums of kernels or generator functions indexed by continuous parameters that control local and global features such as their translation, dilation, modulation and shape. L\'{e}vy random fields and their stochastic integrals are employed to induce prior distributions for the unknown functions or, equivalently, for the number of kernels and for the parameters governing their features. Scaling, shape, and other features of the generating functions are location-specific to allow quite different function properties in different parts of the space, as with wavelet bases and other methods employing overcomplete dictionaries. We provide conditions under which the stochastic expansions converge in specified Besov or Sobolev norms. Under a Gaussian error model, this may be viewed as a sparse regression problem, with regularization induced via the L\'{e}vy random field prior distribution. Posterior inference for the unknown functions is based on a reversible jump Markov chain Monte Carlo algorithm. We compare the L\'{e}vy Adaptive Regression Kernel (LARK) method to wavelet-based methods using some of the standard test functions, and illustrate its flexibility and adaptability in nonstationary applications.Comment: Published in at http://dx.doi.org/10.1214/11-AOS889 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Learning Theory and Approximation

The main goal of this workshop – the third one of this type at the MFO – has been to blend mathematical results from statistical learning theory and approximation theory to strengthen both disciplines and use synergistic effects to work on current research questions. Learning theory aims at modeling unknown function relations and data structures from samples in an automatic manner. Approximation theory is naturally used for the advancement and closely connected to the further development of learning theory, in particular for the exploration of new useful algorithms, and for the theoretical understanding of existing methods. Conversely, the study of learning theory also gives rise to interesting theoretical problems for approximation theory such as the approximation and sparse representation of functions or the construction of rich kernel reproducing Hilbert spaces on general metric spaces. This workshop has concentrated on the following recent topics: Pitchfork bifurcation of dynamical systems arising from mathematical foundations of cell development; regularized kernel based learning in the Big Data situation; deep learning; convergence rates of learning and online learning algorithms; numerical refinement algorithms to learning; statistical robustness of regularized kernel based learning

Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis

The positivity of the energy in relativistic quantum mechanics implies that wave functions can be continued analytically to the forward tube T in complex spacetime. For Klein-Gordon particles, we interpret T as an extended (8D) classical phase space containing all 6D classical phase spaces as symplectic submanifolds. The evaluation maps $e_z: f\to f(z)$ of wave functions on T are relativistic coherent states reducing to the Gaussian coherent states in the nonrelativistic limit. It is known that no covariant probability interpretation exists for Klein-Gordon particles in real spacetime because the time component of the conserved "probability current" can attain negative values even for positive-energy solutions. We show that this problem is solved very naturally in complex spacetime, where $|f(x-iy)|^2$ is interpreted as a probability density on all 6D phase spaces in T which, when integrated over the "momentum" variables y, gives a conserved spacetime probability current whose time component is a positive regularization of the usual one. Similar results are obtained for Dirac particles, where the evaluation maps $e_z$ are spinor-valued relativistic coherent states. For free quantized Klein-Gordon and Dirac fields, the above formalism extends to n-particle/antiparticle coherent states whose scalar products are Wightman functions. The 2-point function plays the role of a reproducing kernel for the one-particle and antiparticle subspaces.Comment: 252 pages, no figures. Originally published as a book by North-Holland, 1990. Reviewed by Robert Hermann in Bulletin of the AMS Vol. 28 #1, January 1993, pp. 130-132; see http://wavelets.co

Large Scale Kernel Methods for Fun and Profit

Kernel methods are among the most flexible classes of machine learning models with strong theoretical guarantees. Wide classes of functions can be approximated arbitrarily well with kernels, while fast convergence and learning rates have been formally shown to hold. Exact kernel methods are known to scale poorly with increasing dataset size, and we believe that one of the factors limiting their usage in modern machine learning is the lack of scalable and easy to use algorithms and software. The main goal of this thesis is to study kernel methods from the point of view of efficient learning, with particular emphasis on large-scale data, but also on low-latency training, and user efficiency. We improve the state-of-the-art for scaling kernel solvers to datasets with billions of points using the Falkon algorithm, which combines random projections with fast optimization. Running it on GPUs, we show how to fully utilize available computing power for training kernel machines. To boost the ease-of-use of approximate kernel solvers, we propose an algorithm for automated hyperparameter tuning. By minimizing a penalized loss function, a model can be learned together with its hyperparameters, reducing the time needed for user-driven experimentation. In the setting of multi-class learning, we show that – under stringent but realistic assumptions on the separation between classes – a wide set of algorithms needs much fewer data points than in the more general setting (without assumptions on class separation) to reach the same accuracy. The first part of the thesis develops a framework for efficient and scalable kernel machines. This raises the question of whether our approaches can be used successfully in real-world applications, especially compared to alternatives based on deep learning which are often deemed hard to beat. The second part aims to investigate this question on two main applications, chosen because of the paramount importance of having an efficient algorithm. First, we consider the problem of instance segmentation of images taken from the iCub robot. Here Falkon is used as part of a larger pipeline, but the efficiency afforded by our solver is essential to ensure smooth human-robot interactions. In the second instance, we consider time-series forecasting of wind speed, analysing the relevance of different physical variables on the predictions themselves. We investigate different schemes to adapt i.i.d. learning to the time-series setting. Overall, this work aims to demonstrate, through novel algorithms and examples, that kernel methods are up to computationally demanding tasks, and that there are concrete applications in which their use is warranted and more efficient than that of other, more complex, and less theoretically grounded models

Toeplitz Operators on Semi-Simple Lie Groups

Let $G/K$ be a Hermitian symmetric space of non-compact type. We consider for the so-called minimal Olshanskii semigroup $\Gamma\subset G^C$, the C$^*$-algebra $T$ generated by all Toeplitz operators $T_f$ on the Hardy space $H^2(\Gamma)\subset L^2(G)$. We describe the construction of ideals of $T$ associated to boundary strata of the domain $\Gamma$