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If you work in a quantitative or technical field, there is little doubt that you or your team has worked long and hard to define which metrics to measure and track. Using data-driven metrics is a critical practice for making rational decisions and deciphering truth in a complex and noisy world. However, as others have pointed out, an over-reliance on metrics can lead to poor outcomes. In an effort…
Attention is a type of layer in a neural network that is widely regarded to be one of the most important breakthroughs that enabled the development of modern AI systems and large language models. At its heart, attention is a mechanism for explicitly drawing relationships between items in a set. In natural language processing, the set being processed are words (or tokens) and attention enables the…
Euler’s number $e := 2.71828\dots$ has, to me, always been a semi-mysterious number. While I understood many facts about $e$, I never felt I ever truly understood what it really was – it’s core essence so to speak. I believe that part of the reason for my confusion is that $e$ is often taught coming from two seemingly different perspectives: Either it is introduced in the context of compound…
If you’re a practitioner of machine learning, then there is little doubt you have seen or used an algorithm that falls into the general category of kernel methods. The premier example of such methods is the support vector machine. When introduced to these algorithms, one is taught that one must provide the algorithm with a kernel function that, intuitively, computes a degree of “similarity”…
The dot product is a fundamental operation on two Euclidean vectors that captures a notion of similarity between the vectors. In this post, we’ll define the dot product and offer a number of angles for which to intuit the idea captured by this fundamental operation.
Cells are crowded spaces packed with biomolecules colliding and interacting with one another. Despite this chaotic environment, biologists routinely describe intracellular functions using the clean mathematical language of networks. In this post I will attempt to reconcile these two seemingly contradictory perspectives of the cell. This post will serve as a first part in a series of blog posts I…
I am currently reading Sun Tzu’s Art of War and am finding much wisdom in it. I have been taking notes during my reading and I thought I’d share them in this post. Here I cover Books 1 and 2.
In Part 1 of this series, we introduced the denoising diffusion probabilistic model for modeling and sampling from complex distributions. We described the diffusion model as a model that can generate new samples by learning how to reverse a diffusion process. In this post, we provide more theoretical justification for the objective function used to fit diffusion models and make connections between…
Diffusion models are a family of state-of-the-art probabilistic generative models that have achieved ground breaking results in a number of fields ranging from image generation to protein structure design. In Part 1 of this two-part series, I will walk through the denoising diffusion probabilistic model (DDPM) as presented by Ho, Jain, and Abbeel (2020). Specifically, we will walk through the…
We human beings use our vision as our chief sense for understanding the world, and thus when we are confronted with data, we try to understand that data through visualization. Dimensionality reduction methods, such as PCA, t-SNE, and UMAP, are approaches designed to enable the visualization of high-dimensional data. Unfortunately, because these methods inevitably distort aspects of the data, these…