Chi-Ken Lu

5d transition metal oxides, such as iridates, have attracted significant interest in condensed matter physics throughout the past decade owing to their fascinating physical properties that arise from intrinsically strong spin-orbit coupling (SOC) and its interplay with other interactions of comparable energy scales. Among the rich family of iridates, iridium dioxide (IrO₂), a simple binary compound long known as a promising catalyst for water splitting, has recently been demonstrated to possess novel topological states and exotic transport properties. The strong SOC and the nonsymmorphic symmetry that IrO₂ possesses introduce symmetry-protected Dirac nodal lines (DNLs) within its band structure as well as a large spin Hall effect in the transport. Here, we review recent advances pertaining to the study of this unique SOC oxide, with an emphasis on the understanding of the topological electronic structures, syntheses of high crystalline quality nanostructures, and experimental measurements of its fundamental transport properties. In particular, the theoretical origin of the presence of the fourfold degenerate DNLs in band structure and its implications in the angle-resolved photoemission spectroscopy measurement and in the spin Hall effect are discussed. We further introduce a variety of synthesis techniques to achieve IrO₂ nanostructures, such as epitaxial thin films and single crystalline nanowires, with the goal of understanding the roles that each key parameter plays in the growth process. Finally, we review the electrical, spin, and thermal transport studies. The transport properties under variable temperatures and magnetic fields reveal themselves to be uniquely sensitive and modifiable by strain, dimensionality (bulk, thin film, nanowire), quantum confinement, film texture, and disorder. The sensitivity, stemming from the competing energy scales of SOC, disorder, and other interactions, enables the creation of a variety of intriguing quantum states of matter.

The analytic inference, e.g., predictive distribution being in closed form, may be an appealing benefit for machine learning practitioners when they treat wide neural networks as Gaussian process in a Bayesian setting. The realistic widths, however, are finite and cause weak deviation from the Gaussianity under which partial marginalization of random variables in a model is straightforward. On the basis of multivariate Edgeworth expansion, we propose a non-Gaussian distribution in differential form to model a finite set of outputs from a random neural network, and derive the corresponding marginal and conditional properties. Thus, we are able to derive the non-Gaussian posterior distribution in Bayesian regression task. In addition, in the bottlenecked deep neural networks, a weight space representation of a deep Gaussian process, the non-Gaussianity is investigated through the marginal kernel and the accompanying small parameters.

The analytic inference, e.g. predictive distribution being in closed form, may be an appealing benefit for machine learning practitioners when they treat wide neural networks as Gaussian process in Bayesian setting. The realistic widths, however, are finite and cause weak deviation from the Gaussianity under which partial marginalization of random variables in a model is straightforward. On the basis of multivariate Edgeworth expansion, we propose a non-Gaussian distribution in differential form to model a finite set of outputs from a random neural network, and derive the corresponding marginal and conditional properties. Thus, we are able to derive the non-Gaussian posterior distribution in Bayesian regression task. In addition, in the bottlenecked deep neural networks, a weight space representation of deep Gaussian process, the non-Gaussianity is investigated through the marginal kernel.

Deep Gaussian Process (DGP) as a model prior in Bayesian learning intuitively exploits the expressive power in function composition. DGPs also offer diverse modeling capabilities, but inference is challenging because marginalization in latent function space is not tractable. With Bochner's theorem, DGP with squared exponential kernel can be viewed as a deep trigonometric network consisting of the random feature layers, sine and cosine activation units, and random weight layers. In the wide limit with a bottleneck, we show that the weight space view yields the same effective covariance functions which were obtained previously in function space. Also, varying the prior distributions over network parameters is equivalent to employing different kernels. As such, DGPs can be translated into the deep bottlenecked trig networks, with which the exact maximum a posteriori estimation can be obtained. Interestingly, the network representation enables the study of DGP's neural tangent kernel, which may also reveal the mean of the intractable predictive distribution. Statistically, unlike the shallow networks, deep networks of finite width have covariance deviating from the limiting kernel, and the inner and outer widths may play different roles in feature learning. Numerical simulations are present to support our findings.

Deep Gaussian Processes (DGPs) were proposed as an expressive Bayesian model capable of a mathematically grounded estimation of uncertainty. The expressivity of DPGs results from not only the compositional character but the distribution propagation within the hierarchy. Recently, it was pointed out that the hierarchical structure of DGP well suited modeling the multi-fidelity regression, in which one is provided sparse observations with high precision and plenty of low fidelity observations. We propose the conditional DGP model in which the latent GPs are directly supported by the fixed lower fidelity data. Then the moment matching method is applied to approximate the marginal prior of conditional DGP with a GP. The obtained effective kernels are implicit functions of the lower-fidelity data, manifesting the expressivity contributed by distribution propagation within the hierarchy. The hyperparameters are learned via optimizing the approximate marginal likelihood. Experiments with synthetic and high dimensional data show comparable performance against other multi-fidelity regression methods, variational inference, and multi-output GP. We conclude that, with the low fidelity data and the hierarchical DGP structure, the effective kernel encodes the inductive bias for true function allowing the compositional freedom.

It is desirable to combine the expressive power of deep learning with Gaussian Process (GP) in one expressive Bayesian learning model. Deep kernel learning showed success as a deep network used for feature extraction. Then, a GP was used as the function model. Recently, it was suggested that, albeit training with marginal likelihood, the deterministic nature of a feature extractor might lead to overfitting, and replacement with a Bayesian network seemed to cure it. Here, we propose the conditional deep Gaussian process (DGP) in which the intermediate GPs in hierarchical composition are supported by the hyperdata and the exposed GP remains zero mean. Motivated by the inducing points in sparse GP, the hyperdata also play the role of function supports, but are hyperparameters rather than random variables. It follows our previous moment matching approach to approximate the marginal prior for conditional DGP with a GP carrying an effective kernel. Thus, as in empirical Bayes, the hyperdata are learned by optimizing the approximate marginal likelihood which implicitly depends on the hyperdata via the kernel. We show the equivalence with the deep kernel learning in the limit of dense hyperdata in latent space. However, the conditional DGP and the corresponding approximate inference enjoy the benefit of being more Bayesian than deep kernel learning. Preliminary extrapolation results demonstrate expressive power from the depth of hierarchy by exploiting the exact covariance and hyperdata learning, in comparison with GP kernel composition, DGP variational inference and deep kernel learning. We also address the non-Gaussian aspect of our model as well as way of upgrading to a full Bayes inference.

Deep Gaussian Processes (DGPs) were proposed as an expressive Bayesian model capable of a mathematically grounded estimation of uncertainty. The expressivity of DPGs results from not only the compositional character but the distribution propagation within the hierarchy. Recently, [1] pointed out that the hierarchical structure of DGP well suited modeling the multi-fidelity regression, in which one is provided sparse observations with high precision and plenty of low fidelity observations. We propose the conditional DGP model in which the latent GPs are directly supported by the fixed lower fidelity data. Then the moment matching method in [2] is applied to approximate the marginal prior of conditional DGP with a GP. The obtained effective kernels are implicit functions of the lower-fidelity data, manifesting the expressivity contributed by distribution propagation within the hierarchy. The hyperparameters are learned via optimizing the approximate marginal likelihood. Experiments with synthetic and high dimensional data show comparable performance against other multi-fidelity regression methods, variational inference, and multi-output GP. We conclude that, with the low fidelity data and the hierarchical DGP structure, the effective kernel encodes the inductive bias for true function allowing the compositional freedom discussed in [3,4].

We propose a standing-wave-decomposition (SWD) approximation to Gaussian process (GP) regression. GP involves a costly matrix inversion operation, which limits applicability to large data analysis. For an input space that can be approximated by a grid and when correlations among data are short ranged, the kernel matrix inversion can be replaced by analytic diagonalization using the SWD. We show that this approach applies to uni- and multidimensional input data, extends to include longer-range correlations, and the grid can be in a latent space and used as inducing points. Through simulations, we show that our approximate method applied to the squared exponential kernel outperforms existing methods in predictive accuracy per unit time in the regime where data are plentiful. Our SWD-GP is recommended for regression analyses where there is a relatively large amount of data and/or there are constraints on computation time.