Mikhail Belkin, Ph.D. - US grants

CAREER: Geometry and High-Dimensional Inference

Mikhail Belkin
Ohio State University

Machine learning techniques for high-dimensional inference are becoming progressively more important as many sources of abundant data ranging from MRI, medical imaging and biological data to sensor networks and to more traditional speech and computer vision data become avail- able and require automated processing. This project will address theoretical and algorithmic issues surrounding manifold and geometric methods for high- dimensional inference.

Intellectual Merit: Three of the fundamental challenges for modern machine learning can be summarized as follows:
. High dimensionality of the data.
. Complex nonlinear structures in the data.
. A large amount of data obtained from modern data sources is unlabeled.
A promising line of research, known as Manifold Learning, emerged in recent years as a way to use certain geometric ideas to construct compact low- dimensional representation of the data and to use unlabeled data for learning. These algorithms have now been successfully used for a variety of applications from motion segmentation to Markov decision processes.
However, our theoretical understanding of these methods is still in its infancy. The main focus of this project is to develop a theoretical framework for analysis of algorithms utilizing geometry of high-dimensional data. Such a framework will bring together techniques from computer science, statistics and mathematics to gain insight into properties of real-world data.
This framework will provide guidelines for designing better algorithms for existing problems as well as extending existing methods to new domains, such as analysis complex output spaces and time dependent data. The PI also plans to investigate usefulness of these ideas in Computer Vision.

Broader impacts: This project aims to build a theoretical foundation for a new class of inference algorithms as well as to design new algorithms for high- dimensional inference and to consider its application. A rigorous theoretical understanding of unlabeled data and its use in learning tasks is likely to have a significant impact in algorithms design and in applications of machine learning techniques in practice. This project will provide research and education opportunities for graduate and undergraduate students, and acquaint researchers from other areas and industry with recent developments and encourage collaborations through interdisciplinary workshops and a Machine Learning school.

http://www.cse.ohio-state.edu/~mbelkin/nsfcareerresearch

This award provides support to students and other young researchers for travel and accomodation to the 2009 Chicago Learning School/Workshop on Theory and Practice of Computational Machine Learning to be held in Chicago, June 1-11, 2009. Tutorial presentations of the Summer School are being recorded and are being made available over the Web. The workshop and summer school provide a forum on geometry and high-dimensional inference in computational learning, and will be effective in bringing these subjects to the attention of a broad scientific community interested in problems of machine learning and inference.

The objective of this program is to develop room-temperature terahertz semiconductor Raman lasers based on giant Raman nonlinearities associated with intersubband transitions in semiconductor superlattices.
The new devices are expected to operate at room temperature because the Raman gain does require population inversion across the low-energy terahertz transition. Giant values of Raman nonlinearity will allow using midinfrared quantum cascade laser butt-joined to the Raman section for pumping. Such device configuration will result in millimeter-sized electrically-pumped terahertz semiconductor laser sources.

The intellectual merit of this research is to explore a novel and highly promising approach to produce the first room-temperature terahertz semiconductor laser source. This work will enhance our understanding of optical nonlinearities and intersubband electron dynamics in semiconductor nanostructures. This project promises significant advances in state of the art terahertz sources, yielding compact semiconductor devices operating at room temperature, at higher power, and with new functionalities such as broadband electric tunability.

The broader impacts are also significant. The proposed research lies at the intersection of the optoelectronics, nonlinear optics, and physics of semiconductors. This combination of disciplines offers a unique educational environment for the students involved in the project.
Knowledge and techniques developed during research will be incorporated into graduate- and undergraduate-level courses, disseminated through publications, technology transfer, and the research groups? websites.
Room-temperature terahertz semiconductor lasers developed as a result of this program are expected to transform existing terahertz instrumentation with applications ranging from high-resolution spectroscopy and local oscillators for radio astronomy to terahertz remote sensing and imaging.

Data analysis is a fundamental problem in computational science, ubiquitous in a broad range of application fields, from computer graphics to geographics information system, from sensor networks to social networks, and from economics to biological science. Two complementary fields that have driven modern data analysis are computational geometry and statistical learning. The former focuses on detailed and precise models characterizing low-dimensional geometric phenomena. The latter focuses on robust or predictive inference of models given noisy high-dimensional data. This project aims to initiate a dialog between these two fields with geometry being the central theme. A closer interaction between them will benefit and advance both fields, and can potentially fundamentally change the way we view and perform data analysis.

Specifically, on one hand, the type of data common in the learning community poses several challenges for traditional computational geometry methods. The shift of focus to these challenges and the modeling of uncertainty central in statistical learning can broaden the scope of computational geometry, and lead to geometric algorithms and models that are more robust to noise and extend to high-dimensional data analysis. On the other hand, computational geometry has developed many elegant structures that contain often detailed and precise information about the underlying domain. Models parameterized using these structures can lead to statistical learning models and algorithms that are richer and more interpretable but remain robust to noise and are predictive.

This project is multi-disciplinary in nature, and will involve fields including computational geometry, algorithms, statistics, differential geometry and topology. Education will be integrated in this project.

The objective of this research is to study both theoretically and experimentally the mechanisms and limitations of rapid electrical control of the emission frequency in single-mode quantum cascade lasers using bias control of the modal effective refractive index. Rapidly tunable single-mode quantum cascade lasers with all-electrical control of the emission frequency will be used to demonstrate a prototype of a free-space optical frequency-modulation link.
Intellectual Merits
The proposed research is aimed at further exploration of the unique physics and design flexibility of quantum cascade lasers. We will study a delicate interplay between the effectiveness of voltage control of the devices frequency and their performance characteristics. This work will create knowledge needed of the realization of mid-infrared semiconductor lasers that can be frequency modulated at speeds well above 1 GHz.
Broader Impacts
The proposed research contains elements of many disciplines, including quantum mechanics, optics, high-frequency electronics, and semiconductor microfabrication and provides an excellent educational environment for graduate and undergraduate students. Knowledge developed during the research will be incorporated into graduate-level courses taught by the principle investigators, and disseminated to the research community through publications, technology transfer, and the research group websites. The devices developed over the course of this project are expected to enable the creation of high-fidelity optical communication links based on frequency-modulation in the 8-12 microns atmospheric transparency window.

Obtaining information from data is one of the most fundamental problems of modern science and technology. The aim of machine learning is to develop algorithms to automatically extract useful information from complex, high-dimensional data. Making progress toward this aim requires developing an understanding of the aspects of data, which are amenable to analysis and can be learned using computationally efficient methods. In particular, modeling non-linear structures in high-dimensional data has become one of the very challenging and active lines of research, which has seen significant progress over the last ten years.

The goal of this project is to develop and analyze new mathematical representations for data, based on spectral and algebraic methods. We will explore how different structures in the data, such as cluster, manifold or parametric model structures, are reflected in their spectral and algebraic properties and how they can be extracted algorithmically from data, paying particular attention to the issues of high dimensionality and non-linearity. These insights will be used to build better and more adaptive algorithms for inference and data analysis tasks.

We will also analyze experimentally and theoretically properties of these algorithms, when data deviates from the posited model structure. This is a key issue in practical applications, which nearly always involve uncertainty and noise.

Proposal ECCS 1150071

Project Abstract

Objective
The research objective of this proposal is the investigation and development of widely and continuously tunable terahertz quantum-cascade lasers and other active devices based upon monolithic and voltage tunable metamaterial waveguides. These waveguides are based on the transmission-line metamaterial concept in which the dispersion relation is tuned by electrical control of integrated capacitive elements. Goals include demonstration of widely tunable terahertz laser sources in various configurations, including tunable distributed feedback lasers, tunable master-oscillator power-amplifiers, and beam-steerable phased-array THz laser emitters.

Intellectual Merit
The intellectual merit is the integration of tunable metamaterial concepts with laser waveguides, in which propagation is sensitively dependent on values of embedded circuit elements. This approach can achieve a much wider range of tunability than can be achieved using temperature or other electrical techniques, with a high degree of monolithic integration. The demonstration of amplifier and distributed feedback configurations will allow the output power to be scaled to several milliwatts or greater with a directive beam pattern.

Broader Impacts
The broader impacts lie in the development of a new approach for achieving tunable semiconductor lasers. Additionally, terahertz technology requires improved sources and detectors for sensing, imaging, and spectroscopy. The core of the educational plan is to enhance undergraduate research within the Electrical Engineering department through the development of a coordinated publicity, outreach, and training program, as well as directly employing undergraduates in this research. Multiple campus resources will be leveraged to attract and fund underrepresented minority students to this research and engineering in general.

Modern machine learning algorithms often encounter a trade off between scalability and modeling power. For the problem of data clustering, Bayesian approaches enjoy numerous modeling advantages over classical methods, but hard clustering methods such as k-means are often preferred in practice due to their simplicity and scalability.

This project explores bridging the gap between classical hard clustering methods and clustering models based on Bayesian nonparametrics. The first step is an asymptotic result connecting the Dirichlet process Gaussian mixture model with a k-means-like algorithm that does not fix the number of clusters in advance. Using this key result, the PI and his team will explore four related research directions which collectively demonstrate the utility of this asymptotic approach: (1) extensions of the analysis to hierarchical Bayesian models, leading to scalable hard clustering methods over multiple data sets; (2) connections to spectral methods and graph clustering, leading to novel and flexible graph clustering methods; (3) extensions beyond the Gaussian setting, leading to new approaches to topic modeling and other discrete-data clustering problems; and (4) extensive experiments in both the computer vision and text domains.

Given that k-means is truly a workhorse of machine learning, these four directions have the potential to impact a wide array of large-scale applications including computer vision, bioinformatics, social network analysis, and many other domains. Furthermore, the research will benefit the broader community through released software and integration into coursework at Ohio State University.

The objective of this EAGER proposal is to design, theoretically model, and experimentally demonstrate a novel class of large-area ultra-thin metasurfaces based on intersubband polaritons with giant nonlinear optical response, orders of magnitude larger than any device reported so far in the scientific literature. Proof-of-concept demonstration efforts will focus on development highly-efficient second harmonic generation mirrors in the mid-infrared regime. The proposed concept is, however, much broader and can be readily extended to other nonlinear optical phenomena.
Intellectual Merits: The proposed research will introduce novel concepts in nonlinear optics and in metamaterial technology. For the first time, nonlinear response of intersubband excitations strongly coupled to optical modes in metal-dielectric-metal microcavities will be investigated. Already giant intersubband optical nonlinearities in coupled-quantum-wells systems can be further boosted by coupling intersubband transitions with ad-hoc engineered resonant modes in metal-dielectric-metal metamaterial cavities.
Broader Impacts: The proposed research combines elements of metamaterial theory, optoelectronics, nonlinear optics, and semiconductor physics. This combination offers a rich, interdisciplinary educational environment for graduate and undergraduate students. Knowledge and techniques developed during our research efforts will be incorporated into focused graduate-level courses currently taught by the PI and Co-PI. The metasurfaces developed in this project are expected to find numerous industrial applications. The PI will continue his annual participation in the NNIN REU program with students involved in the proposed research and involve the co-PI in this effort. At the K-12 level, PI and co-PI will collaborate with NSF-sponsored UTeachEngineering program.

Data analysis is ubiquitous in a broad range of application fields, from computer graphics to geographic information systems, from sensor networks to social networks, from economics to medicine. It represents a fundamental problem in computational science. The project will advance the theoretical understanding of fundamental issues behind data analysis, and develop practical algorithms that will be useful for a broad range of problems in science and engineering.

The project addresses the fundamental problem of reconstructing structure of probability distributions from sampled data. It will investigate the use of tensor-based and other higher order methods, in particular those that allow for efficient optimization. The project lies at the interface of theoretical computer science, machine learning, signal processing and statistics and will have potential impact in all of these fields. In recent years there has been a resurgence of interest in tensor methods in data analysis and inference, particularly in theoretical computer science. These methods will prove useful in a variety of applications in machine learning, signal processing and other fields.

The project will develop algorithms for solving a range of problems including blind source separation, spectral clustering, inference in mixture models and estimating geometry of distributions. It will analyze the complexity of these and related problems. In particular, it will strive to understand the computational efficiency  and dependence on the dimension of the space, studying "the curses and blessings of dimensionality". It will also address a somewhat mysterious discrepancy between sample and algorithmic complexity in our understanding of many high dimensional inference problems.

The results of this work will be disseminated to the broad scientific community through publications in journals, conferences and presentations in various venues, including tutorials. The goals of this project include to implement the practical algorithms and to make the software available online. The research results will also be incorporated in the curriculum of graduate classes taught by the PI and the co-PI. Graduate students supported by this project will receive extensive training in theory, algorithm development and applications.

The scientific objective of this proposal is to create terahertz (THz) frequency combs based on quantum cascade laser sources operating at room-temperature. The research will focus on innovative designs of the quantum cascade laser to produce hundreds of evenly-spaced THz emission lines. Based on these laser sources, the PI intends to demonstrate a prototype spectroscopic system using the frequency-comb emission lines. The proposed devices could lead to room-temperature spectroscopic instrumentation for security and industrial applications based on mass-producible semiconductor components. This project is a multidisciplinary activity that offers a unique educational environment for graduate and undergraduate students involved in the project. At least one graduate student and several undergraduate students will be involved in this research program. The PI will endeavor to recruit these students from underrepresented minorities and will interact with students at the K-12 level through the NSF-sponsored UTeachEningeering program.


The objective of this proposal is to attain the necessary knowledge to create room-temperature quantum cascade lasers having THz frequency combs spanning the entire 0.5-5 THz range. The research will focus on creating quantum cascade laser active region design that provides broadband mid-infrared laser gain, giant third-order optical nonlinearity for efficient mid-infrared frequency comb generation, and giant second-order nonlinearity for efficient down-conversion of mid-infrared frequency comb to THz spectral range. The laser waveguide will be designed to enable low group velocity dispersion for mid-infrared TM00 laser modes and efficient terahertz radiation generation and out-coupling via Cherenkov difference-frequency generation. A subtle interplay of three- and four-wave mixing processes in mid-infrared quantum cascade lasers will be investigated, linewidth measurements of terahertz frequency comb emission from the lasers will be performed, and novel low group-velocity-dispersion quantum cascade laser waveguide designs will be optimized via iterative design-fabrication-measurements process. A prototype frequency comb spectroscopic system based on quantum cascade laser sources and a heterodyne detector, such as room-temperature Schottky-diode or cryogenically-cooled superconducting or hot-electron bolometer, is to be demonstrated.

The goal of machine learning is to extract useful information from data. While the amount of data available to researchers for analysis is ever increasing, much of the data are unlabeled, meaning that the data come without labels indicating their associations with specific learning tasks. Thus understanding unsupervised inference is one of the key problems in machine learning. In addition, data annotated for a certain task may be difficult to use even for tasks only slightly different. This is known as the problem of transfer learning in the literature. To make the most of the available information, machine learning algorithms need to to obtain, analyze and use realistic structural assumptions about the data based on rigorous mathematical models. The proposed work offers students working on this project an opportunity to be exposed to a broad spectrum of topics including machine learning, statistics, geometry and applied mathematics. Students will learn a combination of theory and algorithm development skills in machine learning and data analysis. The results of this work will be disseminated to the broad scientific community through publications in journals, conferences, presentations in various venues, including tutorials and course notes. The material related to this project will be incorporated in PI?s and co-PI's courses. The PIs will also create summer research and practice opportunities for interested undergraduate students in research related to the project.

In this EAGER project an exploration of two types of structural assumptions on the data will be started. Geometric structures in data will be explored, such as hierarchical structure of clusters and density. The use of partial orders for non-geometric data will be explored, based on probabilistic models for partial rankings an orders for problems such as zero-shot learning and transfer learning. By approaching the problem of inference from data within these frameworks, output of this project will be a stepping stone to the challenges of machine learning and to developing efficient algorithms to advance the state-of-the-art both in theory and practice. It is argued argue that these models and the proposed mathematical/algorithmic machinery are amenable to theoretical analysis and will provide insight into properties of real data. Results from the proposed work will broaden the scope of machine learning methods to analyze more complex data in a theoretically well-founded manner.

Support for the 2016 International Quantum Cascade Laser School and Workshop
Nontechnical description: This grant provides funding support to young scientists (postdocs and graduate students) from US institutions with the top-ranked accepted presentation to attend the 2016 International Quantum Cascade Laser School and Workshop (IQCLSW). The IQCLSW conference is held every 2 years since 2004. It serves as a premier venue for all the groups working in the field of quantum cascade lasers for discussing the latest developments in the field and generating new research directions. Quantum cascade lasers are semiconductor lasers operating in mid-infrared and terahertz range of electromagnetic spectrum which have numerous important applications in the fields of chemical and biological sensing, free-space optical communications, defensive infrared countermeasures, securing screening, and remote detection of dangerous substances. The 2016 IQCLSW conference will be held in Cambridge, United Kingdom on September 4-9, 2016. An important component is the 2-day school portion, in which hour-long tutorial presentations are aimed at graduate students and young scientists to rapidly acquaint them with the fundamentals of this rapidly developing technology and thus provide education and training of the new generation of scientists. The 2016 IQCLSW conference will have a significant applications focus and will include and engage mid-infrared and terahertz photonics industry.

Technical description: The 2016 IQCLSW meeting starts with the two-day School section, with 9 hour-long tutorial presentations by top scientists and engineers in the field aimed at students, postdocs, and young scientists. Tutorial presentations are designed to teach the fundamentals of the rapidly evolving field of mid-infrared and terahertz quantum cascade lasers and mid-infrared and terahertz photonics applications. The 3-day Workshop follows the School and will have 4 keynote, 10-15 invited and 45-55 contributed talks, as well as poster sessions, focused on the most recent developments in quantum cascade lasers and their applications. Applications that will be discussed include: high-resolution spectroscopy, microscopy, chemical and biomedical sensing, coherent detection, imaging, security screening, and infrared countermeasures.

This project is to develop a materials platform for mid-infrared integrated photonics and create mid-infrared photonic crystal sensors with parts per billion sensitivity. Such a platform and sensors do not currently exist. High sensitivity will be achieved by targeting strong fundamental vibrational absorption lines of chemical components in the mid-infrared spectral range, also known as the molecular fingerprint region, and employing the principle of slow light in slotted photonic crystal waveguides. The photonic crystal waveguides will be monolithically integrated with mid-infrared quantum cascade lasers and detectors using epitaxial transfer techniques to produce highly compact integrated-photonic chemical sensors operating in the mid-infrared spectral region for a wide range of applications, such as chemical and biomedical sensing, environmental monitoring, and security applications.

The scientific objective of this project to investigate a mid-infrared photonic waveguideing platform spanning molecular fingerprint region (3-14 micrometer) and develop integrated-photonic sensors in the molecular fingerprint region. Silicon-on-insulator devices cannot be operated beyond 3.5 micrometer wavelength range owing to mid-infrared absorption in SiO2, silicon-on-sapphire platform is limited to wavelengths below 5 micrometer owing to sapphire absorption, and silicon itself is only transparent down to 7-8 micrometers. The lack of a suitable photonic waveguiding platform currently limits the realization of a portable absorption spectrometer surrogate of benchtop infrared spectroscopic systems that can effectively probe the molecular fingerprint region below 1500 cm-1. Epitaxially-grown GaAs/AlGaAs waveguides offer a well-established lattice-matched, very low defect density and high index contrast platform for low loss optical circuits in the entire 3-14 micrometer wavelength range. This project aims at developing a GaAs/AlGaAs integrated photonics platform for on-chip mid-infrared photonics spanning the entire mid-infrared spectral range, including the molecular fingerprint region, and using this platform to demonstrate high sensitivity low parts per billion on-chip absorption sensing in the molecular fingerprint region. High sensitivity will be achieved using slotted photonic crystal waveguide devices. The proposed platform is expected to have a detection limit of 0.3 parts per billion for CO2 and 1 part per billion for toluene at their respective peak absorbance wavelengths at 4.23 micrometer and at 13.8 micrometer, respectively.

This PFI: AIR Technology Translation project focuses on translating the technology of semiconductor quantum cascade lasers to fill the need of compact and inexpensive widely-tunable room-temperature sources of terahertz radiation. Terahertz quantum cascade lasers are expected to become the first frequency-agile continuous-wave room-temperature terahertz sources with the cost, mass-producibility potential, size, and operation simplicity of diode lasers. Commercial applications that are expected to benefit from this source technology include: real-time non-contact thickness measurements of coatings in industrial environments (wet and dry paints, plastics, paper, tablet coatings), gas sensing, THz imaging, spectroscopy, and microscopy. The new technology is expected to dramatically reduce the cost and size of existing terahertz systems.

This project addresses the need to increase the power output and simplify processing of continuous-wave room-temperature terahertz quantum cascade laser sources as they translate from research discovery toward commercial application. This will be achieved by using high-quality materials growth, thermal packaging, and innovations in the device active region and waveguide designs. These innovations are expected to boost the power output from the current record of 14 microwatt to over 50 microwatt and simplify device processing. In addition, personnel involved in this project, two graduate students in the Electrical Engineering program and one student in the Master's of Science in Technology Commercialization program, will receive a unique experience of advancing a complex semiconductor device technology to applications.

The project focuses on the development and marketing of frequency-agile narrow-linewidth continuous-wave sources in 1-6 THz spectral range with over 50 microwatt power output, which is sufficient for most terahertz applications. Frequency-agile continuous-wave THz sources are highly desired for THz applications because (a) their narrowband emission frequencies can be selected to fall between atmospheric water absorption lines, (b) they can be used as local oscillator sources for heterodyne THz detectors, high-resolution spectroscopy, and gas sensing, and (c) their radiation can be detected using highly-sensitive room-temperature Schottky-diode based detectors. Commercial applications that are expected to benefit from our source technology include: real-time non-contact thickness measurements of coatings in industrial environments (wet and dry paints, plastics, paper, tablet coatings), gas sensing, THz imaging, and a variety of scientific instruments for materials characterization, such as near-field and far-field THz microscopes, high-resolution spectrometers, and local oscillator sources for radio-astronomy. The basic device technology is intracavity difference-frequency generation in mid-infrared quantum cascade lasers. To achieve high-power continuous-wave operation goal the team will combine their expertise in growth and thermal packaging of high-power continuous-wave mid-infrared quantum cascade lasers and will further implement innovations in the device active region and waveguide that are expected to boost room-temperature continuous-wave power output from the current record of 14 microwatts to over 50 microwatts and simplify device processing.

The scientific objective of this proposal is to develop and test artificial semiconductor nonlinear optical materials and semiconductor quantum cascade lasers based on indium-aluminum-gallium-nitride materials. The indium-aluminum-gallium-nitride materials system has fundamental advantages over the materials that were previously used for making quantum cascade lasers and artificial semiconductor nonlinear optical materials. In particular, indium-aluminum-gallium-nitride semiconductor lasers operating in the terahertz spectral range (frequencies in the range of 1-10 THz) are expected to be able to operate at room temperature, unlike semiconductor lasers previously demonstrated in other materials systems. Room-temperature terahertz semiconductor lasers will have a major transformative impact on the instrumentation operating in this frequency range. Indium-aluminum-gallium-nitride materials are also expected to enable the creation of a novel kind of nonlinear metamaterials for operation at the wavelengths used by fiber-optics telecommunication equipment with sub-1-picosecond response time. Two graduate students will be trained during the course of the program. The two principal investigators will also continue their annual participation in the National Science Foundation research experience for undergraduate program and in various K-12 outreach activities at their institutions.

Technical Description.
The objective of this proposal is to develop intersubband optoelectronic devices based on strain-compensated InGaN/AlGaN/GaN heterostructures grown on non-polar m-plane GaN substrates for operation in the short-wavelength infrared (wavelengths in the range 1.4-3 microns) and terahertz (wavelengths in the range 30-300 microns) regions of the electromagnetic spectrum. Current intersubband devices rely on materials with relatively low conduction band offsets (<1 eV) and low longitudinal optical phonon energies (~30-40 meV) that, respectively, prevent intersubband devices from operating in the short-wavelength infrared and limit the operation of terahertz quantum cascade lasers to cryogenic temperatures. GaN/AlGaN heterostructures grown on c-plane substrates have been previously investigated to overcome the abovementioned problems. GaN-based materials system offers conduction band offsets over 2 eV and have optical phonon energies of ~90 meV. However, strain-dependent piezo-electric fields make it virtually impossible to produce desired intersubband bandstructure in practical devices grown on c-plane substrates. Additionally, relatively small heterostructure thickness, limited by strain, and poor optical field confinement in the heterostructure prevented efficient light-matter interaction in devices reported previously. The proposed AlInGaN heterostructures on m-plane GaN substrates are free from strain-induced fields making reliable intersubband bandstructure design possible. Strain-compensation will be used to overcome critical thickness constrains in materials growth. The heterostructures will be further processed into double-metal plasmonic cavities using photoelectrochemical etching for substrate removal to enable efficient light-matter integration. Two types of intersubband devices will be investigated: double-metal waveguide THz QCLs and intersubband nonlinear metasurfaces for operation in the telecommunication spectral range. The former devices represent a viable path towards developing the first room-temperature electrically pumped semiconductor lasers in the THz spectral range, while the latter devices offer a path for developing intersubband metasurfaces with a giant nonlinear response for short-wavelength infrared.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Science, medicine, business, and engineering are increasingly data-driven. Hypotheses, diagnoses, decisions, and designs are made by gathering and analyzing a wealth of data in search of meaningful patterns. It is the goal of the field of machine learning to develop methods for inferring and reasoning about patterns in the data. The research supported by this award addresses the question of learning from unlabeled data, one of the more vexing problems of data science. The PI's analyze and develop algorithms for problems such as clustering, i.e., finding groups of similar objects, as well as understanding continuously changing attributes in data. By integrating machine learning, modeling and geometric data analysis, this work injects new ideas and methodologies to modern data analysis, helps build practical algorithms for unsupervised and unsupervised learning and analyze their properties and domains of applicability. Students working on this project have a unique opportunity to be exposed to a broad spectrum of topics including machine learning, statistics, geometry and applied mathematics.

On a more technical level, the unifying perspective for the proposed research is that many of these unsupervised learning problems can be viewed as recovering structure or invariants of the underlying continuous space through the lens of the discrete data. This work takes that point of view to consider a number of important aspects of unsupervised learning including hierarchical clustering in the density model, data quantization, graphon clustering and estimation, as well as learning metric structure from data. The project also considers applications of these ideas to supervised learning, particularly in helping to scale algorithms to large data. While the work on this project concentrates on theoretical analyses, these are developed with a view toward practical algorithms, implementations and applications.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.