Ml on Omanshu Thapliyal

So does your Car fail? (A Small Harness for Mathematical Exploration) - Part 3

Wed, 16 Sep 2026 12:02:49 -0400

Part 1:https://omanshuthapliyal.github.io/blog/hrt-conjecture/

Part 2: https://omanshuthapliyal.github.io/blog/hrt-conjecture-part-2/

In the (last couple) (of posts) I explained what the recently (dis-)proven HRT conjecture states, and what it implies. We looked at how a computer-assisted argument can turn a difficult mathematical question into a finite collection of things that can be checked. Here, we try the more mundane inverse problem: how do we set up a small AI harness such that an LLM can suggest where to look, while the things that matter are still checked deterministically? Here’s a speedy recap. The original HRT conjecture asked whether this could ever happen.¹ For a function $$g$$, define the time-frequency shift $$ \pi(a,b)g(x) = e^{2\pi i b x}g(x-a). $$ Given finitely many distinct points $$ \Lambda = \{(a_k,b_k)\}_{k=1}^{N} \subset \mathbb{R}^{2}, $$ HRT asked whether the collection $$\{\pi(a_k,b_k)g\}_{k=1}^{N}$$ is always linearly independent whenever $$g\neq 0$$. Equivalently, can we find nonzero coefficients $$c_k$$ such that $$ \sum_{k=1}^{N} c_k \pi(a_k,b_k)g = 0? $$

Contraction can Help Safety Classifiers keep their Promise

Wed, 12 Aug 2026 01:02:49 -0400

The code for this project can be found at: https://github.com/omanshuthapliyal/contraction_constrained-safety-head. And the paper can be found at: https://arxiv.org/pdf/2610.02853.

A safety classifier can tell us that a prompt looks harmful. That is useful. But it is perhaps not the question we should stop at. Suppose a classifier sees a prompt and says that it is safe. Now suppose we change the prompt slightly. Perhaps we substitute a few words. Perhaps we phrase the same request differently. Perhaps, more abstractly, we nudge the token embeddings of the prompt by a very small amount. Would the classifier still say safe?

Can LLM Finetuning be Stateful? Small states for Large Models - Part 2

Fri, 07 Aug 2026 11:57:21 -0700

The code for this project can be found at: https://github.com/omanshuthapliyal/HRM-adapter. And the paper can be found at: https://arxiv.org/pdf/2606.26290.

In the previous post, I introduced the basic idea behind a stateful adapter. Instead of asking a small trainable module to only modify the current transformer representation, we allow it to maintain a state: $$ s_t = \bar{A}s_{t-1} + \bar{B}h_t, $$ and use that state to generate a residual correction, $$ r_t = Cs_t. $$ The important unresolved question was: if this adapter gets a small amount of memory, what should it remember? The answer I want to explore is the same one that appeared in the discussion of Hankel singular values:

Can LLM Finetuning be Stateful? Small states for Large Models - Part 1

Fri, 31 Jul 2026 11:57:21 -0700

In the last post I talked about Hankel operators of simple linear dynamical systems. Such operators, and either eigenvalues, “can be used as a proxy for how much memory a task actually requires in a state-space model”. And later we tied their input-output relations empirically tie the memory requirements of a post-trained state space model (SSM) to its Hankel singular value (HSV) rates. Further, HSV-based balanced truncation can offer us to selectively choose ‘some’ eigenvalues to keep that involve higher input-to-output signal strength over time. And so, we used Hankel singular values to decide which states of a dynamical system are worth keeping. However, a language model also processes a sequence of tokens over time. We saw task-dependent HSV decay rates for various tasks for different SSM-based models. Similarly, could we give the model a small state, one deliberately built around that same notion of “worth keeping”, without retraining the entire model?

What Hankel Spectra Reveal About Memory

Tue, 26 May 2026 11:09:02 -0700

Suppose we are given a linear dynamical system $$G=(A,B,C,D)$$ (recall that a linear system is an input $$x$$ to output $$y$$ map such that $$y_k = Cx_k + Du_k, x_{k+1}=Ax_k+Bu_k$$), and $$G$$ has some dimension $$d$$. Often times one has to find an “equivalent” system $$\hat{G}$$– one that has an input-output behavior as close to $$G$$ as possible, while ideally having a lower dimension $$\hat{d}$$. This very vital demand from the new system $$\hat{G}$$ to model the original (higher dimensional) system as closely as possible has many applications: building smaller plants for controller synthesis, reduced order system modeling, compressing fluid or other PDE-derived dynamical systems into manageable surrogate models, etc.

Why are Gaussian distributions present Everywhere?

Sun, 11 Jan 2026 11:09:02 -0700

Well, if one were to cut to the chase, it appears to be because of the Central Limit Theorem (CLT). However, there is more underlying structure under the hood as to why the Gaussians seem to be the ‘chosen’ distributions. In this post we will go through some in-depth analysis beyond the typical textbook to try to get some juice worth the squeeze. And hopefully it would demystify the ubiquity – from statistics, to thermodynamics, to measurement errors – of Gauss’s eponymous distributions. To this end, this write up is a shallow dive into Gaussian ubiquity from a few different angles.

State Space Models: How Control Theory can echo Machine Learning

Mon, 06 Oct 2025 18:57:21 -0700

I wrote in a previous post how transformers are kind of like system identification methods applied to some sequence in a state space. In this post we try to understand if the other way round is true.

We already saw previously ¹ how predicting words/tokens is a sequence to sequence mapping problem, which is the subject of a lot of prediction, estimation, and controls problems. In fact, we leveraged a transformer’s sequence mapping ability and applied it to a controls problem of trajectory prediction for a simple 2D car model ¹. Can we do the other way round? Turns out: yes we can!

How Transformers Echo Control Theory

Sat, 27 Sep 2025 11:57:21 -0700

Transformer models have captivated a lot of AI research in most of the past decade, and in this post my goal is to make them seem more interesting to controls people, who have much to contribute to the field. I will be making gross generalizations to draw such entry points. At least, I found transformer models more interesting after finding a few parallels in dynamical systems & controls ¹. This post attempts to explore those parallels.

Reward is enough — when can we 'reinforce' the learning?

Wed, 13 Aug 2025 11:07:46 -0700

This blog post summarizes the papers Settling the Reward Hypothesis¹ and Utility Theory for Sequential Decision Making ².

The reward hypothesis is at the core of Reinforcement Learning (RL) in that whether sequential action problems can be posed as some sort of a reward maximization. One of the founders of modern RL, Richard S. Sutton, puts it as³:

“That all of what we mean by goals and purposes can be well thought of as maximization of the expected value of the cumulative sum of a received scalar signal (reward).”

Technology Readiness Levels of AI methods

Tue, 15 Oct 2024 11:07:46 -0700

Recent news of the SpaceX catching its Starship Super Heavy booster¹ is quite discussed and marveled upon in the media, and rightly so. Executing what was done can perhaps be explained as trying to catch a falling stick, then balancing it vertically, all while the stick is over 200 ft tall, weighs over 150 tonnes, spews fire, and costs over tens of millions of dollars. One cannot but feel a little giddy as a controls engineering enthusiast over this massive success. It makes me think why we are closer to automation in certain domains/problems than others? Why, for example, is your vacuum cleaning robot seemingly better at navigating in a cluttered environment than your possibly self-driving car? Why is the tremendous progress in machine learning & AI not permeating outside of our web lives as much as we expected in the early 2000’s?

MLP Approximation

Thu, 12 Sep 2019 10:27:01 -0400

Almost always we hear about classification or machine learning problems, the go-to methods to solve the problem are neural networks, or multi-layered percetrons (MLP). Now function approximation problems, which is what classification is, are very well defined in terms of consistency, accuracy, and other abilities of the approximator. Why are MLPs then able to approximate functions well? Especially given the fact that most of the problems in coming up with candidate architectures, activation functions, etc. are ‘design problems’.

ML SoTA Resources

Sun, 18 Aug 2019 22:36:00 +0530

The internet is filled with machine learning resources, and one of the most annoying things about them is the sheer volume. There are many attempts at making compilations of papers, code, and current status quo in the vastly active, and fast paced field. This post is to serve as a collection of my own where I will only post state of the art in machine learning.

Most of these links are updated frequently and maintained by the community, and I have collected these only for reference. I am not trying to create a new standard, but I will try to update the list if and when I come around doing that.