Colloquia and Seminars

Classical approaches to geometric differential invariants begin by specifying a transformation group, analytically deriving the corresponding invariant differential quantities, and constructing numerical schemes to approximate them. This work takes a data-driven perspective and introduces a self-supervised deep learning framework for learning group-equivariant semi-local geometric operators directly from point samples of planar curves. Specifically, we train a neural network to act as a discrete differentiation operator. Remarkably, the learned operator behaves as a discrete approximation to differentiation with respect to the corresponding group-invariant arc length. The framework requires no fixed parameterization and remains robust under non-uniform sampling and noise. Experiments demonstrate that the learned operators generalize across a wide range of curve geometries and successfully handle line-preserving transformation groups, including the Euclidean, similarity and equi-affine groups. The learned differential operators transfer directly to downstream shape-analysis tasks without retraining. The resulting framework is a data-driven foundation for approximating geometric invariants from discrete observations.

Free recall is a standard memory task in which a subject is presented with a sequence of items and, after presentation, asked to retrieve as many as possible in any order. Understanding free recall is not only a question of retrieval capacity, but also of retrieval structure: recalled items appear in clusters, transitions are highly non-random, and retrieval depends strongly on temporal position. We introduce the Tree of Memory (TOM), a computational model in which memory is represented by a hierarchical episodic data structure (episodic tree) augmented with a sparse semantic graph. Experience is organized across multiple temporal scales in different levels of the hierarchy, and recall is modeled as a probabilistic search process over this representation. Specifically, retrieval proceeds through a stochastic depth-first traversal of the episodic tree, interleaved with local exploration of semantic edges whenever an item is reached.

We characterize distinct asymptotic recall regimes and show that the expected number of retrieved items grows logarithmically or as a sublinear power law, depending on how the probability of traversing deeper levels of the memory hierarchy scales with list length. We also study how the same memory representation supports different retrieval algorithms. This allows us to ask how changes in search strategy and auxiliary structure affect recall efficiency. One variant introduces additional semantic links that guide retrieval through a learned sequence, yielding linear recall in list length with high probability. Another uses a small set of pointers to recently formed subtrees, enabling efficient retrieval of entire temporal chunks. These variants highlight explicit trade-offs between recall performance, representation complexity, scalability, and auxiliary memory. Simulations further show that TOM reproduces canonical empirical signatures of free recall. Together, these results suggest that hierarchical representations and probabilistic graph search provide a compact computational framework for understanding both spontaneous and strategy-driven memory retrieval.

This talk investigates how external uncertainty estimates and expert specialization can enhance neural time-series forecasting. Across two complementary studies, we examine the mechanisms underlying these improvements, how design choices shape them, and the conditions that limit their effectiveness.

The first study asks whether quantile forecasts from an external predictor can guide a diffusion-based probabilistic forecaster. We investigate how different forms of quantile integration shape forecasting performance and find that residual conditioning throughout the network yields the most consistent improvements on the primary backbone. The resulting method, Control-TS, employs a trainable controller that injects quantile-derived corrections into a frozen diffusion forecaster, improving its probabilistic accuracy across all eight evaluated benchmarks. We also introduce the Conditional Linear Predictive Score (CLPS), which assesses the downstream forecasting utility of generated futures using a linear model.

The second study disentangles expert specialization from model capacity and routing in multivariate forecasting. Controlled comparisons show that informative assignments of observed channels improve forecasting beyond matched shared capacity and random allocation, while the benefit of input-dependent routing varies across backbones. Synthetic experiments reveal how persistent channel information supports specialization when short observation windows conceal predictive differences. Motivated by these findings, we develop a plug-and-play framework that augments a frozen pretrained forecaster with low-rank residual experts. Alternating expert learning with closed-form estimation of channel-specific mixture weights tailors these corrections to individual channels while preserving the original predictor.

Planning electricity supply, anticipating traffic, investing in the Stock market, and assessing how a patient’s health may evolve all involve uncertainty about the future. This seminar presents two complementary approaches to improving time-series forecasting through structured guidance and expert specialization. The first study investigates predicted quantiles as distributional controls for diffusion forecasting. Quantile forecasts summarize uncertainty at individual future time steps, while diffusion forecasters represent it through sampled future trajectories. We systematically investigate guidance mechanisms and identify residual control throughout the denoising network as the most consistently effective approach. We introduce Control-TS, which learns a quantile-control pathway while keeping the pretrained backbone frozen. Across eight benchmarks, Control-TS improves the backbone’s probabilistic accuracy, achieves competitive performance against state-of-the-art forecasting models, and retains gains in capacity-matched comparisons. Further analyses examine the roles of quantile granularity, predictor choice, and forecast quality. We also introduce the Conditional Linear Predictive Score (CLPS) to assess the downstream forecasting utility of generated futures.

The second contribution disentangles the roles of capacity, parameter sharing, and routing in multivariate time-series forecasting. We introduce GalaxyTS, an expressive synthetic dataset with controllable predictive structure and known optimal forecasts, revealing when specialization helps and how channels should share experts. These insights motivate an identity-only router that introduces no additional routing parameters and improves on the learned identity-only baseline. Our plug-and-play framework combines this router with low-rank residual experts to adapt pretrained forecasters, recovering their original predictions exactly when corrections are disabled. Evaluation across multiple architectures, forecast horizons, and established benchmarks demonstrates accuracy gains over the original forecasters and competitive performance with low computational overhead.

Brownian Bridge Diffusion Models (BBDM) offer an appealing approach to image restoration by constructing a stochastic bridge between a clean image and its degraded observation. Although the bridge schedule plays an important role in reconstruction quality, it is typically chosen heuristically, motivating a principled framework for schedule design.

In this talk, an analytical framework for understanding how BBDM schedules affect the distribution of reconstructed images will be presented. Under a Mixture-of-Gaussians prior, we obtain a closed-form posterior and an ideal minimum mean squared error (MMSE) denoiser. We then compare the posterior law with a tractable surrogate for the BBDM reconstruction law. The surrogate preserves the posterior mean but exhibits a covariance deficit, revealing a coordinate-wise tradeoff between reconstruction fidelity, measured by mean squared error, and distributional accuracy, measured by Wasserstein distance.

Building on this analysis, we derive complementary schedule-design objectives and propose schedules based on analytical bounds that are independent of the prior and degradation. Extensive experiments on controlled MoG settings confirm full alignment between theory and practice, and experiments on the FFHQ dataset across inpainting, deblurring, and super-resolution tasks validate the practical value of our schedule-design criteria.

Rollups improve blockchain scalability by aggregating transactions, yet sequencer centralization introduces risks like ordering manipulation and censorship, which are difficult to verify. While current approaches focus on distributed sequencing or state correctness, they overlook the need for ordering accountability. We introduce DOPPIO, a framework for retrospective verification of fair sequencing. DOPPIO employs a dual-commitment scheme, binding the sequencer to both the ordering metadata and execution batch, and extends the base detector with value-weighted and spatial-concentration variants that, aggregated over consecutive batches, detect sustained localized maximal extractable value (MEV) attacks. We characterize the per-pair limit of observer-based detection: sub-margin delay masking of an isolated transaction pair is indistinguishable from network jitter. Validated on real Ethereum mainnet mempool data, DOPPIO provides accountability with negligible communication overhead and no added execution latency.