Research Archive

We worked with the largest community of sustainable businesses in the city to turn their sustainability goals into a data-driven efficiency roadmap. We identified a single operational change yielding 30k in YoY utilities savings using a combination of smart utility meters, solar cells, and space usage data.

Deep learning models learn internal representations of the world by progressively compressing nuisance variation while preserving task-relevant structure. We study this process through the lens of statistical sufficiency and group actions. Given a compact group acting on the sample space, we show that when the likelihood or conditional target law is invariant along group orbits, the orbit quotient X/G is not only an invariant representation but a sufficient statistic for the underlying parameters; under a likelihood-ratio separation condition it is minimally sufficient. This connects classical sufficiency theory to geometric factorization of nuisance symmetries. We then specialize to compact Lie groups and homogeneous spaces, where Peter-Weyl theory provides harmonic coordinates. For finite-band harmonic exponential families, we prove that empirical generalized Fourier coefficients form minimally sufficient statistics under standard full-rank conditions. Using Clebsch-Gordan decomposition, we express the normalization constant of these families as the trivial-irreducible projection of an exponential tensor algebra, yielding an algebraic view of the partition function. Finally, we extend minimal sufficiency from single tasks to families of tasks by characterizing a task-complete representation as the quotient by the intersection of task-specific invariance subgroups. This framework links invariant and equivariant deep learning, harmonic analysis, and statistical sufficiency, and offers a structural perspective on representation minimality, AI safety, and interpretability.

We formulate machine unlearning for online L-BFGS as a counterfactual state-alignment problem. Given an actual event stream and a deletion-edited counterfactual stream, the target of unlearning is the optimizer state that would have arisen had the deleted samples never been processed. We introduce state-aware metrics that separately measure parameter error, memory-operator error, combined state error, and update-direction error. The memory metric compares the inverse-Hessian actions induced by the o-L-BFGS memory, rather than treating curvature pairs as of finite influence. Under convexity assumptions, we derive a recursive bound on counterfactual state deviation. We then evaluate a state-aware benchmark of deletion interventions, including memory-only and parameter-only corrections, against an counterfactual oracle model. These results show that unlearning for online L-BFGS is not merely a parameter-correction problem: it requires alignment with a realizable counterfactual optimizer state.

Individuals have the right to be forgotten, but does AI has the right to forget?

In this work, we introduce a localized, MDL-based divergence measure that quantifies the structural complexity of induced subgraphs relative to a global reference model. The measure compares the compressibility of local neighborhoods under globally fitted and locally optimized comparator models while penalizing model flexibility, yielding a statistically grounded notion of local structural surprise. The results show that the measure converges under controlled ablations, is robust to sampling choices, and detects meaningful structural irregularities that are invariant to geometric embedding. This framework generalizes MDL-based network analysis to arbitrary information graphs and provides a principled bridge between global structure and agent-level experience.

The Lab investigates the math behind a machine's right to forget, and the human's right to be forgotten as presented by the seminal Sekhari paper.

We propose an MDL-based measure of network structural complexity. The method utilizes the two-phase MDL approach for describing random and nonrandom variation.

Unlearning is more than performance. It's a matter of model state.

We review Sinha and Mahajan's take on the sufficiency of an information state. We're such fans, we decide to take it even further.

A primer on entropy, mutual information, and variational inference.

Continuing our Foundations series, we explore sigma-algebras—the mathematical scaffolding that makes probability rigorous and connects abstract theory to real-world randomness in our models.