Date/Time : 11 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Heng Yang, Harvard University

Many fundamental estimation and decision-making problems in robotics are formulated as non-convex mathematical optimization problems. Decades of research have produced numerous algorithms to efficiently solve these problems, yet offering few performance guarantees. In this talk, I aim to advocate using convex semidefinite relaxations to design a new paradigm of algorithms that work out of the box and offer strong performance certificates.

I first briefly introduce the celebrated Lasserre’s hierarchy of moment and sums-of-squares (SOS) relaxations, which relaxes polynomial optimization and decision problems as a hierarchy of convex semidefinite programs (SDP). I then present three applications of this hierarchy in robotics. In the first application, we apply the hierarchy to solve outlier-robust estimation problems in robot perception and show that SDP relaxations provide certificates of optimality. In the second application, we combine this hierarchy with conformal prediction from statistics and set-membership estimation from control theory and demonstrate such relaxations provide certificates of uncertainty. In the third application, we revisit trajectory optimization problems arising in nonlinear optimal control and show that, by exploiting correlative sparsity, SDP relaxations produce near-optimal control trajectories. I end the talk by discussing computational challenges of this framework in our applications and opportunities to tackle them.

Add to calendar :

Date/Time : 12 Oct 2023 14:00
Venue : S16 #03-04
Speaker : Wang Peng, University of Michigan

Over the past decade, deep learning has proven to be a highly effective method for extracting meaningful features from high-dimensional data. This work attempts to unveil the mystery of feature learning in deep networks. Specifically, for a multi-class classification problem, we explore how the features of training data evolve across the intermediate layers of a trained neural network. We investigate this problem based on simple deep linear networks trained on linearly separable data, and we analyze how the output features in each layer concentrate around the means of their respective classes. Remarkably, when the deep linear network is trained using gradient descent from a small orthogonal initialization, we theoretically prove that the features exhibit a linear decay in the measure of within-class feature variability as we move from shallow to deep layers. Moreover, our extensive experiments not only validate our theoretical findings numerically but also reveal a similar pattern in deep nonlinear networks which well aligns with recent empirical studies.

Add to calendar :

Date/Time : 10 Oct 2023 10:30
Venue : S17 #04-06 (SR1)
Speaker : Charles Cheung, NVIDIA AI Technology Center

In this talk, I will talk about a few directions and use cases of AI for Science. PINNs and neural operator are used for solving many engineering problems that involves differential equations. We will walk through the basic concept of PINNs and neural operator and introduce NVIDIA modulus, an SDK for training PINNs and neural operator. We will also share a recent project about transfer learning.

Add to calendar :

Date/Time : 22 Aug 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Shahar Mendelson, Australian National University

Consider an isotropic measure \mu on R^d (i.e., centred and whose covariance is the identity) and let X_1,…,X_m be independent, selected according to \mu. If \Gamma=m^{-1/2} \sum_{i=1}^m <X_i,->e_i is the random operator whose rows are X_i/\sqrt{m}, how does the random set \Gamma S^{d-1} typically look like? For example, if the extremal singular values of \Gamma are close to 1 then \Gamma S^{d-1} is “well approximated” by a d-dimensional section of S^{m-1} and vice-versa. But is it possible to give a more accurate description of that set?

I will show that under minimal assumptions on \mu, with high probability and uniformly in t \in S^{d-1}, each vector \Gamma t inherits the structure of the one-dimensional marginal <X,t> in a strong sense. If time permits I will also present a few generalisations of this fact – for an arbitrary indexing set.

Add to calendar :

Date/Time : 13 Apr 2023 16:00
Venue : S16 #03-06
Speaker : Karthyek Murthy, Singapore University of Technology and Design

Several problems in data-driven decision-making and risk management suffer from data imbalance, a term referring to settings where a small fraction of data has an outsized impact on estimating one or more decision-making criteria. Due to the paucity of relevant samples, such problems are usually approached with the “estimate, then optimize” workflow involving a model estimation from data in the first step before plugging in the trained model to solve various downstream optimization tasks. As biases due to model selection, misspecification, and overfitting to in-sample data are especially difficult to avoid in the first-step estimated model in settings affected by data imbalance, we construct novel locally robust optimization formulations in which the first-step estimation has no effect, locally, on the optimal solutions obtained.

We show that this local insensitivity translates to improved out-of-sample performance freed from the first-order impact of model errors introduced in the first-step estimation. A key ingredient in achieving this local robustness is a novel debiasing procedure that adds a non-parametric bias correction term to the objective. The debiased formulation retains convexity, and the imputation of the correction term relies only on a non-restrictive large deviations behavior conducive for transferring knowledge from representative data-rich regions to the data-scarce tail regions suffering from imbalance. The bias correction gets determined by the extent of model error in the estimation step and the specifics of the stochastic program in the optimization step, thereby serving as a scalable “smart-correction” step bridging the disparate goals in estimation and optimization.

Add to calendar :

Date/Time : 13 Apr 2023 10:00
Venue : via Zoom
Speaker : Daniel Sanz-Alonso, University of Chicago

Join Zoom Meeting
https://nus-sg.zoom.us/j/81261210349?pwd=M2VlVjUvR1o4dGVodDY4MnBtRzMzUT09

Meeting ID: 812 6121 0349
Passcode: 557052

Semi-supervised learning refers to the problem of recovering an input-output map using many unlabeled examples and a few labeled ones. In this talk I will survey several mathematical questions arising from the Bayesian formulation of graph-based semi-supervised learning. These questions include the modeling of prior distributions for functions on graphs, the derivation of continuum limits for the posterior, the design of scalable posterior sampling algorithms, and the contraction of the posterior in the large data limit.

Add to calendar :

Date/Time : 30 Mar 2023 10:00
Venue : via Zoom
Speaker : Xun Huan, University of Michigan

Join Zoom Meeting
https://nus-sg.zoom.us/j/9775606531?pwd=cVZ6eTRYS3ZzVUlFNXRuSlhMVjJSUT09

Meeting ID: 977 560 6531
Passcode: 123321

Experiments are indispensable for learning and developing models in engineering and science. A careful design of these often-expensive data-acquisition opportunities can be immensely beneficial. Simulation-based optimal experimental design, while leveraging a predictive model, offers a framework to systematically quantify and maximize the value of experiments. We focus on designing a finite sequence of experiments, seeking optimal design policies that can (a) adapt to newly collected data during the sequence (i.e. feedback) and (b) anticipate future changes (i.e. lookahead). We cast this sequential decision-making problem in a Bayesian setting with information-based utilities, and solve it numerically via policy gradient methods from reinforcement learning. In particular, we directly parameterize the policies and value functions by neural networks—thus adopting an actor-critic approach—and improve them using gradient estimates produced from simulated design and observation sequences. We demonstrate the overall method on a problem of optimal sensor movement for contaminant source inversion.

Add to calendar :

Date/Time : 16 Feb 2023 16:00
Venue : S16 #05-98
Speaker : Xia Kelin, Nanyang Technological University

Artificial intelligence (AI) based molecular data analysis has begun to gain momentum due to the great advancement in experimental data, computational power and learning models. However, a major issue that remains for all AI-based learning models is the efficient molecular representations and featurization. Here we propose advanced mathematics-based molecular representations and featurization (or feature engineering). Molecular structures and their interactions are represented as various simplicial complexes (Rips complex, Neighborhood complex, Dowker complex, and Hom-complex), hypergraphs, and Tor-algebra-based models. Molecular descriptors are systematically generated from various persistent invariants, including persistent homology, persistent Ricci curvature, persistent spectral, and persistent Tor-algebra. These features are combined with machine learning and deep learning models, including random forest, CNN, RNN, Transformer, BERT, and others. They have demonstrated great advantage over traditional models in drug design and material informatics.

Add to calendar :

Date/Time : 06 Jan 2023 14:00
Venue : S17 #05-12 (SR4)
Speaker : Tanya Veeravalli, University of Illinios, Urbana-Champaign

There has been a great deal of recent interest in learning and approximation of functions that can be expressed as expectations of a given nonlinearity with respect to its random internal parameters. Examples of such representations include “infinitely wide” neural nets, where the underlying nonlinearity is given by the activation function of an individual neuron. In this talk, we will bring this perspective to function representation by neural stochastic differential equations (SDEs). A neural SDE is an Itô diffusion process whose drift and diffusion matrix are elements of some parametric families. We will show that the ability of a neural SDE to realize nonlinear functions of its initial condition can be related to the problem of optimally steering a certain deterministic dynamical system between two given points in finite time. This auxiliary system is obtained by formally replacing the Brownian motion in the SDE by a deterministic control input. We derive upper and lower bounds on the minimum control effort needed to accomplish this steering; these bounds may be of independent interest in the context of motion planning and deterministic optimal control.

Add to calendar :