Publications

The null space of the $k$-th order Laplacian $\mathbf{\mathcal L}_k$, known as the $k$-th homology vector space, encodes the non-trivial topology of a manifold or a network. Understanding the structure of the homology embedding can thus disclose geometric or topological information from the data. The study of the null space embedding of the graph Laplacian $\mathbf{\mathcal L}_0$ has spurred new research and applications, such as spectral clustering algorithms with theoretical guarantees and estimators of the Stochastic Block Model. In this work, we investigate the geometry of the $k$-th homology embedding and focus on cases reminiscent of spectral clustering. Namely, we analyze the connected sum of manifolds as a perturbation to the direct sum of their homology embeddings. We propose an algorithm to factorize the homology embedding into subspaces corresponding to a manifold’s simplest topological components. The proposed framework is applied to the shortest homologous loop detection problem, a problem known to be NP-hard in general. Our spectral loop detection algorithm scales better than existing methods and is effective on diverse data such as point clouds and images.

Many social mammals form discrete social communities within larger populations. For nonterritorial, polygynous, size-dimorphic species, sex- and age-class differences in life-history requirements might mediate differences in social connectedness and transitions among communities. We conducted social network analysis and community detection with an extensive data set of 1081 individually identified wild giraffes, Giraffa camelopardalis, over 5 years to test predictions that adult males and young of both sexes show greater social connectedness (degree, closeness and betweenness centrality) and transition more often among social communities than adult females, which form stronger and more stable relationships. We also expected that young animals would be more socially connected than adults. Using both static and dynamic network clustering techniques, we detected four distinct mixed-sex social communities, which we termed ‘super-communities’ to differentiate this apex level of social organization from intermediate-level female-only communities. Most (∼70%) giraffes remained within their same super-community, and those (usually adult males) that visited a different super-community often returned to their original super-community. Males - both adults and calves – had higher social centrality scores than females, and adult males were closer to all other individuals in the network and transitioned among super-communities twice as often as females and calves, reflecting their roaming reproductive strategy of seeking females in oestrus. Of all age and sex classes, young males had the most social ties and highest betweenness (moved most often among groups), which we attributed to social exploration prior to natal dispersal. Overall, female giraffes have stronger social associations than males, but males exceed females in measures of social connectedness, reflecting differences in reproductive and life-history profiles. Our findings suggest that giraffe translocations that do not consider sociality are likely to break up established social associations and potentially reduce fitness.

Optical absorbance is used to study the kinetics of methylammonium lead iodide (MAPbI3) thin film degradation in response to combinations of moisture, oxygen, and illumination over a range of temperatures. 105 degradations were conducted over 41 unique environmental conditions. We discover that water acts synergistically with oxygen in a water-accelerated photo-oxidation (WPO) pathway. This pathway is the dominant pathway at 25 $^\circ$C and is 10, 100, 1000, and >1000 times faster than dry photooxidation (DPO), degradation via hydrate formation, thermal degradation, and blue light degradation, respectively. We find that the rate determining step for DPO is proton abstraction from methylammonium while for WPO it is proton abstraction from water, which occurs at a faster rate and results in water acting as an accelerant for photooxidation of MAPbI3. A full kinetic rate equation is derived and fitted to the data to determine activation energies and rate constants.

The manifold Helmholtzian (1-Laplacian) operator $\Delta_1$ elegantly generalizes the Laplace-Beltrami operator to vector fields on a manifold $\mathcal M$. In this work, we propose the estimation of the manifold Helmholtzian from point cloud data by a weighted 1-Laplacian $\mathbf{\mathcal L}_1$. While higher order Laplacians ave been introduced and studied, this work is the first to present a graph Helmholtzian constructed from a simplicial complex as an estimator for the continuous operator in a non-parametric setting. Equipped with the geometric and topological information about $\mathcal M$, the Helmholtzian is a useful tool for the analysis of flows and vector fields on $\mathcal M$ via the Helmholtz-Hodge theorem. In addition, the $\mathbf{\mathcal L}_1$ allows the smoothing, prediction, and feature extraction of the flows. We demonstrate these possibilities on substantial sets of synthetic and real point cloud datasets with non-trivial topological structures; and provide theoretical results on the limit of $\mathbf{\mathcal L}_1$ to $\Delta_1$.

Many manifold embedding algorithms fail apparently when the data manifold has a large aspect ratio (such as a long, thin strip). Here, we formulate success and failure in terms of finding a smooth embedding, showing also that the problem is pervasive and more complex than previously recognized. Mathematically, success is possible under very broad conditions, provided that embedding is done by carefully selected eigenfunctions of the Laplace-Beltrami operator $\Delta$. Hence, we propose a bicriterial Independent Eigencoordinate Selection (IES) algorithm that selects smooth embeddings with few eigenvectors. The algorithm is grounded in theory, has low computational overhead, and is successful on synthetic and large real data.

One of the aims of both linear and non-linear dimension reduction is to find a reduced set of collective variables that describe the data manifold. While algorithms return abstract coordinates such as spaces spanned by eigenvectors of data-dependent matrices, one can often associate these with features of the data, and hence with domain-related meaning. Usually, finding these domain-related or physical meanings is done via visual inspection by an expert. Our work formulates this problem as sparse, non-parametric, non-linear recovery of the manifold coordinates over a user-defined dictionary of domain-related functions. We show that the original problem can be transformed into a linear Group Lasso problem, and demonstrate the effectiveness of the method on molecular simulation data.

Dynamic extensions of Stochastic block model (SBM) are of importance in several fields that generate temporal interaction data. These models, besides producing compact and interpretable network representations, can be useful in applications such as link prediction or network forecasting. In this paper we present a conditional pseudo-likelihood based extension to dynamic SBM that can be efficiently estimated by optimizing a regularized objective. Our formulation leads to a highly scalable approach that can handle very large networks, even with millions of nodes. We also extend our formalism to causal impact for networks that allows us to quantify the impact of external events on a time dependent sequence of networks. We support our work with extensive results on both synthetic and real networks.

In all manifold learning algorithms and tasks setting the kernel bandwidth $\varepsilon$ used construct the graph Laplacian is critical. We address this problem by choosing a quality criterion for the Laplacian, that measures its ability to preserve the geometry of the data. For this, we exploit the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator. Experiments show that this principled approach is effective and robust.

We demonstrate a process to manipulate and culture human embryonic stem cells with optical tweezers. This method is adopted to study the conditions necessary for successful differentiation and colonization of the stem cells.

Random laser actions in ultraviolet and visible regions have been demonstrated based on the composites consisting of bio-inspired diatom frustules. Owing to the low optical loss derived from porous network of diatom structures, we report wide spectrum range random lasers arising from GaN film and Rh6G dye via using biological diatoms as scattering centers. Interestingly, both ultraviolet and visible-range random laser actions with very sharp peaks can be easily obtained, with the average length of optics cavity closed to the average size of diatom frustules in both cases, indicating the excellent optical confinement of diatom frustules. It is expected that the first proof of concept shown here can pave an avenue toward future broad-range random lasers and eco-friendly biophotonics devices with high performance and wide spectrum response.

A new approach is proposed to light up band-edge stimulated emission arising from a semiconductor with dipole-forbidden band-gap transition. To illustrate our working principle, here we demonstrate the feasibility on the composite of SnO2 nanowires (NWs) and chicken albumen. SnO2 NWs, which merely emit visible defect emission, are observed to generate a strong ultraviolet fluorescence centered at 387 nm assisted by chicken albumen at room temperature. In addition, a stunning laser action is further discovered in the albumen/SnO2 NWs composite system. The underlying mechanism is interpreted in terms of the fluorescence resonance energy transfer (FRET) from the chicken albumen protein to SnO2 NWs. More importantly, the giant oscillator strength of shallow defect states, which is served orders of magnitude larger than that of the free exciton, plays a decisive role. Our approach therefore shows that bio-materials exhibit a great potential in applications for novel light emitters, which may open up a new avenue for the development of bio-inspired optoelectronic devices.