Theses - Dept. of Mathematicshttps://hdl.handle.net/1721.1/76042019-09-22T12:40:34Z2019-09-22T12:40:34ZLarge-scale optimization Methods for data-science applicationsLu, Haihao,Ph.D.Massachusetts Institute of Technology.https://hdl.handle.net/1721.1/1222722019-09-20T03:04:11Z2019-01-01T00:00:00ZLarge-scale optimization Methods for data-science applications
Lu, Haihao,Ph.D.Massachusetts Institute of Technology.
In this thesis, we present several contributions of large scale optimization methods with the applications in data science and machine learning. In the first part, we present new computational methods and associated computational guarantees for solving convex optimization problems using first-order methods. We consider general convex optimization problem, where we presume knowledge of a strict lower bound (like what happened in empirical risk minimization in machine learning). We introduce a new functional measure called the growth constant for the convex objective function, that measures how quickly the level sets grow relative to the function value, and that plays a fundamental role in the complexity analysis. Based on such measure, we present new computational guarantees for both smooth and non-smooth convex optimization, that can improve existing computational guarantees in several ways, most notably when the initial iterate is far from the optimal solution set.; The usual approach to developing and analyzing first-order methods for convex optimization always assumes that either the gradient of the objective function is uniformly continuous (in the smooth setting) or the objective function itself is uniformly continuous. However, in many settings, especially in machine learning applications, the convex function is neither of them. For example, the Poisson Linear Inverse Model, the D-optimal design problem, the Support Vector Machine problem, etc. In the second part, we develop a notion of relative smoothness, relative continuity and relative strong convexity that is determined relative to a user-specified "reference function" (that should be computationally tractable for algorithms), and we show that many differentiable convex functions are relatively smooth or relatively continuous with respect to a correspondingly fairly-simple reference function.; We extend the mirror descent algorithm to our new setting, with associated computational guarantees. Gradient Boosting Machine (GBM) introduced by Friedman is an extremely powerful supervised learning algorithm that is widely used in practice -- it routinely features as a leading algorithm in machine learning competitions such as Kaggle and the KDDCup. In the third part, we propose the Randomized Gradient Boosting Machine (RGBM) and the Accelerated Gradient Boosting Machine (AGBM). RGBM leads to significant computational gains compared to GBM, by using a randomization scheme to reduce the search in the space of weak-learners. AGBM incorporate Nesterov's acceleration techniques into the design of GBM, and this is the first GBM type of algorithm with theoretically-justified accelerated convergence rate. We demonstrate the effectiveness of RGBM and AGBM over GBM in obtaining a model with good training and/or testing data fidelity.
Thesis: Ph. D. in Mathematics and Operations Research, Massachusetts Institute of Technology, Department of Mathematics, 2019; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 203-211).
2019-01-01T00:00:00ZPoint processes of representation theoretic originCuenca, Cesar(Cesar A.)https://hdl.handle.net/1721.1/1221902019-09-19T03:01:52Z2019-01-01T00:00:00ZPoint processes of representation theoretic origin
Cuenca, Cesar(Cesar A.)
There are two parts to this thesis. In the first part we compute the correlation functions of the 4-parameter family of BC type Z-measures. The result is given explicitly in terms of Gauss's hypergeometric function. The BC type Z-measures are point processes on the punctured positive real line. They arise as interpolations of the spectral measures of a distinguished family of spherical representations of certain infinite-dimensional symmetric spaces. In representation-theoretic terms, our result solves the problem of noncommutative harmonic for the aforementioned family of representations. The second part of the text is based on joint work with Grigori Olshanski. We consider a new 5-parameter family of probability measures on the space of infinite point configurations of a discrete lattice. One of the 5 parameters is a quantization parameter and the measures in the family are closely related to the BC type Z-measures. We prove that the new measures serve as orthogonality weights for symmetric function analogues of the multivariate q-Racah polynomials. Further we show that the q-Racah symmetric functions (and their corresponding orthogonality measures) can be degenerated into symmetric function analogues of the big q-Jacobi, q-Meixner and Al-Salam-Carlitz polynomials, thus giving rise to a partial q-Askey scheme hierarchy in the algebra of symmetric functions.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 191-195).
2019-01-01T00:00:00ZHydrodynamic analogues of quantum corrals and Friedel oscillationsCristea-Platon, Tudor.https://hdl.handle.net/1721.1/1221872019-09-18T03:02:26Z2019-01-01T00:00:00ZHydrodynamic analogues of quantum corrals and Friedel oscillations
Cristea-Platon, Tudor.
We consider the walking droplet (or 'walker') system discovered in 2005 by Yves Couder and coworkers. We investigate experimentally and theoretically the behaviour of this hydrodynamic pilot-wave system in both closed and open geometries. First, we consider the dynamics and statistics of walkers confined to corrals. In the elliptical corral, we demonstrate that by introducing a submerged topographical defect, one can create statistical projection effects analogous to the quantum mirage effect arising in quantum corrals. We also report a link between the droplet's statistics and the mean wave field. In the circular corral, we investigate a parameter regime marked by periodic and weakly aperiodic orbits, then characterise the emergence and breakdown of double quantisation, reminiscent of that arising for walker motion in a harmonic potential. In the chaotic regime, we test the theoretical result of Durey et al. relating the walker statistics to the mean wave-field. We also rationalise the striking similarity between this mean wave-field and the circular corral's dominant azimuthally-symmetric Faraday mode. Our corral studies underscore the compatibly of the notion of quantum eigenstates and particle trajectories in closed geometries. We proceed by exploring a new hydrodynamic quantum analogue of the Friedel oscillations arising when a walker interacts with a submerged circular well, which acts as a localised region of high excitability. In so doing, we report the first successful realisation of an open hydrodynamic quantum analogue. We conclude by comparing the hydrodynamic systems to their quantum counterparts. Our work illustrates how, in the closed and open settings considered herein, a pilot-wave dynamics of the form envisaged by de Broglie may lead naturally to emergent statistics similar in form to those predicted by standard quantum mechanics.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 145-153).
2019-01-01T00:00:00ZTotally positive spaces : topology and applicationsGalashin, Pavel(Pavel A.)https://hdl.handle.net/1721.1/1221862019-09-18T03:03:02Z2019-01-01T00:00:00ZTotally positive spaces : topology and applications
Galashin, Pavel(Pavel A.)
This thesis studies topological spaces arising in total positivity. Examples include the totally nonnegative Grassmannian Gr[subscripts >_0](k, n), Lusztig's totally nonnegative part (G/P)[subscripts >_0] of a partial flag variety, Lam's compactification of the space of electrical networks, and the space of (boundary correlation matrices of) planar Ising networks. We show that all these spaces are homeomorphic to closed balls. In addition, we confirm conjectures of Postnikov and Williams that the CW complexes Gr[subscripts >_0](k, n) and (G/P)[subscripts >_0] are regular. This implies that the closure of each positroid cell inside Gr[subscripts >_0](k, n) is homeomorphic to a closed ball. We discuss the close relationship between the above spaces and the physics of scattering amplitudes, which has served as a motivation for most of our results. In the second part of the thesis, we investigate the space of planar Ising networks. We give a simple stratification-preserving homeomorphism between this space and the totally nonnegative orthogonal Grassmannian, describing boundary correlation matrices of the planar Ising model by inequalities. Under our correspondence, Kramers-Wannier's high/low temperature duality transforms into the cyclic symmetry of Gr[subscripts >_0](k, n).
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 195-203).
2019-01-01T00:00:00Z