Palle E. T. Jorgensen, Myung-Sin Song, James Tian
In this paper we show how specific families of positive definite kernels serve as powerful tools in analyses of iteration algorithms for multiple layer feedforward Neural Network models. Our focus is on particular kernels that adapt well to learning algorithms for data-sets/features which display intrinsic self-similarities at feedforward iterations of scaling.
James Tian
This paper develops a nonlinear operator dynamic that progressively removes the influence of a prescribed feature subspace while retaining maximal structure elsewhere. The induced sequence of positive operators is monotone, admits an exact residual decomposition, and converges to the classical shorted operator. Transporting this dynamic to reproducing kernel Hilbert spaces yields a corresponding family of kernels that converges to the largest kernel dominated by the original one and annihilating the given subspace. In the finite-sample setting, the associated Gram operators inherit a structured residual decomposition that leads to a canonical form of kernel ridge regression and a principled way to enforce nuisance invariance. This gives a unified operator-analytic approach to invariant kernel construction and structured regularization in data analysis.
James Tian
We introduce random-kernel networks, a multilayer extension of random feature models where depth is created by deterministic kernel composition and randomness enters only in the outermost layer. We prove that deeper constructions can approximate certain functions with fewer Monte Carlo samples than any shallow counterpart, establishing a depth separation theorem in sample complexity.
Sherwin Varghese, James Tian
A review of over 160,000 customer cases indicates that about 90% of time is spent by the product support for solving around 10% of subset of tickets where a trivial solution may not exist. Many of these challenging cases require the support of several engineers working together within a "swarm", and some also need to go to development support as bugs. These challenging customer issues represent a major opportunity for machine learning and knowledge graph that identifies the ideal engineer / group of engineers(swarm) that can best address the solution, reducing the wait times for the customer. The concrete ML task we consider here is a learning-to-rank(LTR) task that given an incident and a set of engineers currently assigned to the incident (which might be the empty set in the non-swarming context), produce a ranked list of engineers best fit to help resolve that incident. To calculate the rankings, we may consider a wide variety of input features including the incident description provided by the customer, the affected component(s), engineer ratings of their expertise, knowledge base article text written by engineers, response to customer text written by engineers, and historic swarming data. The central hypothesis test is that by including a holistic set of contextual data around which cases an engineer has solved, we can significantly improve the LTR algorithm over benchmark models. The article proposes a novel approach of modelling Knowledge Graph embeddings from multiple data sources, including the swarm information. The results obtained proves that by incorporating this additional context, we can improve the recommendations significantly over traditional machine learning methods like TF-IDF.
Palle E. T. Jorgensen, Myung-Sin Song, James Tian
Motivated by applications, we consider here new operator theoretic approaches to Conditional mean embeddings (CME). Our present results combine a spectral analysis-based optimization scheme with the use of kernels, stochastic processes, and constructive learning algorithms. For initially given non-linear data, we consider optimization-based feature selections. This entails the use of convex sets of positive definite (p.d.) kernels in a construction of optimal feature selection via regression algorithms from learning models. Thus, with initial inputs of training data (for a suitable learning algorithm,) each choice of p.d. kernel $K$ in turn yields a variety of Hilbert spaces and realizations of features. A novel idea here is that we shall allow an optimization over selected sets of kernels $K$ from a convex set $C$ of positive definite kernels $K$. Hence our \textquotedblleft optimal\textquotedblright{} choices of feature representations will depend on a secondary optimization over p.d. kernels $K$ within a specified convex set $C$.