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Category: math

  • On the Possibility of Small Networks for Physics-Informed Learning

    On the Possibility of Small Networks for Physics-Informed Learning A new kind of hyperparameter study The post On the Possibility of Small Networks for Physics-Informed Learning appeared first on Towards Data Science. Conor Rowan Go to original source

  • Implementing Vibe Proving with Reinforcement Learning

    Implementing Vibe Proving with Reinforcement Learning How to make LLMs reason with verifiable, step-by-step logic (Part 2) The post Implementing Vibe Proving with Reinforcement Learning appeared first on Towards Data Science. Jacopo Tagliabue Go to original source

  • Is Your Model Time-Blind? The Case for Cyclical Feature Encoding

    Is Your Model Time-Blind? The Case for Cyclical Feature Encoding How cyclical encoding improves machine learning prediction The post Is Your Model Time-Blind? The Case for Cyclical Feature Encoding appeared first on Towards Data Science. Gustavo Santos Go to original source

  • The Subset Sum Problem Solved in Linear Time for Dense Enough Inputs

    The Subset Sum Problem Solved in Linear Time for Dense Enough Inputs An optimal solution to the well-known NP-complete problem, when the input values are close enough to each other. The post The Subset Sum Problem Solved in Linear Time for Dense Enough Inputs appeared first on Towards Data Science. Tigran Hayrapetyan Go to original…

  • Struggling with Data Science? 5 Common Beginner Mistakes

    Struggling with Data Science? 5 Common Beginner Mistakes Avoid these mistakes to fast track your data science career. The post Struggling with Data Science? 5 Common Beginner Mistakes appeared first on Towards Data Science. Egor Howell Go to original source

  • Learning Triton One Kernel at a Time: Softmax

    Learning Triton One Kernel at a Time: Softmax All you need to know about a fast, readable and PyTorch-ready softmax kernel The post Learning Triton One Kernel at a Time: Softmax appeared first on Towards Data Science. Ryan Pégoud Go to original source

  • How Relevance Models Foreshadowed Transformers for NLP

    How Relevance Models Foreshadowed Transformers for NLP Tracing the history of LLM attention: standing on the shoulders of giants The post How Relevance Models Foreshadowed Transformers for NLP appeared first on Towards Data Science. Sean Moran Go to original source

  • Spearman Correlation Coefficient for When Pearson Isn’t Enough

    Spearman Correlation Coefficient for When Pearson Isn’t Enough Not all relationships are linear, and that is where Spearman comes in. The post Spearman Correlation Coefficient for When Pearson Isn’t Enough appeared first on Towards Data Science. Nikhil Dasari Go to original source

  • We Didn’t Invent Attention — We Just Rediscovered It

    We Didn’t Invent Attention — We Just Rediscovered It How selective amplification emerged across evolution, chemistry, and AI through convergent mathematical solutions The post We Didn’t Invent Attention — We Just Rediscovered It appeared first on Towards Data Science. Javier Marin Go to original source

  • Multiple Linear Regression Explained Simply (Part 1)

    Multiple Linear Regression Explained Simply (Part 1) The math behind fitting a plane instead of a line. The post Multiple Linear Regression Explained Simply (Part 1) appeared first on Towards Data Science. Nikhil Dasari Go to original source

  • Implementing the Fourier Transform Numerically in Python: A Step-by-Step Guide

    Implementing the Fourier Transform Numerically in Python: A Step-by-Step Guide What if the FFT functions in NumPy and SciPy don’t actually compute the Fourier transform you think they do? The post Implementing the Fourier Transform Numerically in Python: A Step-by-Step Guide appeared first on Towards Data Science. JUNIOR JUMBONG Go to original source

  • The Theory of Universal Computation: Bayesian Optimality, Solomonoff Induction & AIXI

    The Theory of Universal Computation: Bayesian Optimality, Solomonoff Induction & AIXI Is it possible to build a perfect induction machine? The post The Theory of Universal Computation: Bayesian Optimality, Solomonoff Induction & AIXI appeared first on Towards Data Science. Angjelin Hila Go to original source

  • Using Python to Build a Calculator

    Using Python to Build a Calculator A beginner-friendly Python project to understand conditional statements, loops and recursive functions The post Using Python to Build a Calculator appeared first on Towards Data Science. Mahnoor Javed Go to original source

  • Implementing the Gaussian Challenge in Python

    Implementing the Gaussian Challenge in Python Beginner-friendly tutorial to understand range function and Python loops The post Implementing the Gaussian Challenge in Python appeared first on Towards Data Science. Mahnoor Javed Go to original source

  • From Tokens to Theorems: Building a Neuro-Symbolic AI Mathematician

    From Tokens to Theorems: Building a Neuro-Symbolic AI Mathematician The next Gauss may not be born — they may be spun up in the cloud The post From Tokens to Theorems: Building a Neuro-Symbolic AI Mathematician appeared first on Towards Data Science. Sean Moran Go to original source

  • The Beauty of Space-Filling Curves: Understanding the Hilbert Curve

    The Beauty of Space-Filling Curves: Understanding the Hilbert Curve A quick journey from theory to implementation and application The post The Beauty of Space-Filling Curves: Understanding the Hilbert Curve appeared first on Towards Data Science. Paul Fröhling Go to original source

  • Stochastic Differential Equations and Temperature — NASA Climate Data pt. 2

    Stochastic Differential Equations and Temperature — NASA Climate Data pt. 2 The Ornstein-Uhlenbeck process in Python The post Stochastic Differential Equations and Temperature — NASA Climate Data pt. 2 appeared first on Towards Data Science. Marco Hening Tallarico Go to original source

  • Understanding Matrices | Part 4: Matrix Inverse

    Understanding Matrices | Part 4: Matrix Inverse The physical meaning of matrix inversion, related formulas, and how inversion behaves on several special types of matrices. The post Understanding Matrices | Part 4: Matrix Inverse appeared first on Towards Data Science. Tigran Hayrapetyan Go to original source

  • The Math You Need to Pan and Tilt 360° Images

    The Math You Need to Pan and Tilt 360° Images Panning a spherical image is just a horizontal roll, but tilting it vertically is much trickier. Let’s see the math! The post The Math You Need to Pan and Tilt 360° Images appeared first on Towards Data Science. Thomas Rouch Go to original source

  • Positional Embeddings in Transformers: A Math Guide to RoPE & ALiBi

    Positional Embeddings in Transformers: A Math Guide to RoPE & ALiBi Learn APE, RoPE, and ALiBi positional embeddings for GPT — intuitions, math, PyTorch code, and experiments on TinyStories The post Positional Embeddings in Transformers: A Math Guide to RoPE & ALiBi appeared first on Towards Data Science. Sathya Krishnan Suresh Go to original source

  • Cracking the Density Code: Why MAF Flows Where KDE Stalls

    Cracking the Density Code: Why MAF Flows Where KDE Stalls Learn why autoregressive flows are the superior density estimation tool for high-dimensional data The post Cracking the Density Code: Why MAF Flows Where KDE Stalls appeared first on Towards Data Science. Zackary Nay Go to original source

  • A Bird’s-Eye View of Linear Algebra: Why Is Matrix Multiplication Like That?

    A Bird’s-Eye View of Linear Algebra: Why Is Matrix Multiplication Like That? Since the way we manipulate high-dimensional vectors is primarily matrix multiplication, it isn’t a stretch to say it is the bedrock of the modern AI revolution. The post A Bird’s-Eye View of Linear Algebra: Why Is Matrix Multiplication Like That? appeared first on…

  • From Rules to Relationships: How Machines Are Learning to Understand Each Other

    From Rules to Relationships: How Machines Are Learning to Understand Each Other Using knowledge graphs to handle the unexpected in semantic communication The post From Rules to Relationships: How Machines Are Learning to Understand Each Other appeared first on Towards Data Science. Shireesh Kumar Singh Go to original source

  • From Equal Weights to Smart Weights: OTPO’s Approach to Better LLM Alignment

    From Equal Weights to Smart Weights: OTPO’s Approach to Better LLM Alignment Using optimal transport to weight what matters most In LLM-generated responses The post From Equal Weights to Smart Weights: OTPO’s Approach to Better LLM Alignment appeared first on Towards Data Science. Sudheer Singh Go to original source

  • POSET Representations in Python Can Have a Huge Impact on Business

    POSET Representations in Python Can Have a Huge Impact on Business Discover how POSET indicators transform data into coherent scoring systems, enabling meaningful comparisons while preserving the data’s multi-dimensional semantic structure. The post POSET Representations in Python Can Have a Huge Impact on Business appeared first on Towards Data Science. Andrea D’Agostino Go to original…

  • Understanding Matrices | Part 2: Matrix-Matrix Multiplication

    Understanding Matrices | Part 2: Matrix-Matrix Multiplication The physical meaning of multiplying two matrices and how it works on several special matrices. The post Understanding Matrices | Part 2: Matrix-Matrix Multiplication appeared first on Towards Data Science. Tigran Hayrapetyan Go to original source

  • Regularisation: A Deep Dive into Theory, Implementation, and Practical Insights

    Regularisation: A Deep Dive into Theory, Implementation, and Practical Insights A detailed guide on controlling overfitting and increasing the stability of your models. The post Regularisation: A Deep Dive into Theory, Implementation, and Practical Insights appeared first on Towards Data Science. Sourav Mohile Go to original source

  • Exploring the Proportional Odds Model for Ordinal Logistic Regression

    Exploring the Proportional Odds Model for Ordinal Logistic Regression Understanding and Implementing Brant’s Tests in Ordinal Logistic Regression with Python The post Exploring the Proportional Odds Model for Ordinal Logistic Regression appeared first on Towards Data Science. JUNIOR JUMBONG Go to original source

  • A Bird’s-Eye View of Linear Algebra: Measure of a Map — Determinants

    A Bird’s-Eye View of Linear Algebra: Measure of a Map — Determinants We roll up our sleeves and start to deal with matrices The post A Bird’s-Eye View of Linear Algebra: Measure of a Map — Determinants appeared first on Towards Data Science. Rohit Pandey Go to original source

  • A Bird’s Eye View of Linear Algebra: The Basics

    A Bird’s Eye View of Linear Algebra: The Basics We think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury. The post A Bird’s Eye View of Linear Algebra: The Basics appeared first on Towards Data Science. Rohit Pandey Go to original source

  • Understanding Matrices | Part 1: Matrix-Vector Multiplication

    Understanding Matrices | Part 1: Matrix-Vector Multiplication The physical meaning of multiplying a matrix by a vector, and how it works on several special matrices. The post Understanding Matrices | Part 1: Matrix-Vector Multiplication appeared first on Towards Data Science. Tigran Hayrapetyan Go to original source

  • Multiple Linear Regression Analysis

    Multiple Linear Regression Analysis Implementation of multiple linear regression on real data: Assumption checks, model evaluation, and interpretation of results using Python. The post Multiple Linear Regression Analysis appeared first on Towards Data Science. JUNIOR JUMBONG Go to original source

  • How to Learn the Math Needed for Machine Learning

    How to Learn the Math Needed for Machine Learning Maths can be a scary topic for people. Many of you want to work in machine learning, but the maths skills needed may seem overwhelming. I am here to tell you that it’s nowhere as intimidating as you may think and to give you a roadmap, resources,…

  • The Total Derivative: Correcting the Misconception of Backpropagation’s Chain Rule

    The Total Derivative: Correcting the Misconception of Backpropagation’s Chain Rule This article uses concepts from this brilliant paper. For a deeper understanding of the mathematics please refer to the paper. Here we try to present the math in a more intuitive and explicit way, with some important nuances highlighted. 1 Introduction Discussions about Backpropagation often…

  • From a Point to L∞

    From a Point to L∞ Why you should read this  As someone who did a Bachelors in Mathematics I was first introduced to L¹ and L² as a measure of Distance… now it seems to be a measure of error — where have we gone wrong? But jokes aside, there seems to be this misconception that L₁ and L₂…

  • Attaining LLM Certainty with AI Decision Circuits

    Attaining LLM Certainty with AI Decision Circuits The promise of AI agents has taken the world by storm. Agents can interact with the world around them, write articles (not this one though), take actions on your behalf, and generally make the difficult part of automating any task easy and approachable.  Agents take aim at the most…

  • Explained: How Does L1 Regularization Perform Feature Selection?

    Explained: How Does L1 Regularization Perform Feature Selection? Feature Selection is the process of selecting an optimal subset of features from a given set of features; an optimal feature subset is the one which maximizes the performance of the model on the given task. Feature selection can be a manual or rather explicit process when…

  • Why CatBoost Works So Well: The Engineering Behind the Magic

    Why CatBoost Works So Well: The Engineering Behind the Magic Gradient boosting is a cornerstone technique for modeling tabular data due to its speed and simplicity. It delivers great results without any fuss. When you look around you’ll see multiple options like LightGBM, XGBoost, etc. Catboost is one such variant. In this post, we will…

  • Unlock the Power of ROC Curves: Intuitive Insights for Better Model Evaluation

    Unlock the Power of ROC Curves: Intuitive Insights for Better Model Evaluation We’ve all been in that moment, right? Staring at a chart as if it’s some ancient script, wondering how we’re supposed to make sense of it all. That’s exactly how I felt when I was asked to explain the AUC for the ROC…

  • How to Optimize your Python Program for Slowness

    How to Optimize your Python Program for Slowness Also available: A Rust version of this article. Everyone talks about making Python programs faster [1, 2, 3], but what if we pursue the opposite goal? Let’s explore how to make them slower — absurdly slower. Along the way, we’ll examine the nature of computation, the role of memory,…

  • Kernel Case Study: Flash Attention

    Kernel Case Study: Flash Attention The attention mechanism is at the core of modern day transformers. But scaling the context window of these transformers was a major challenge, and it still is even though we are in the era of a million tokens + context window (Qwen 2.5 [1]). There are both considerable compute and memory…

  • From Physics to Probability: Hamiltonian Mechanics for Generative Modeling and MCMC

    From Physics to Probability: Hamiltonian Mechanics for Generative Modeling and MCMC Phase space of a nonlinear pendulum. Photo by the author. Hamiltonian mechanics is a way to describe how physical systems, like planets or pendulums, move over time, focusing on energy rather than just forces. By reframing complex dynamics through energy lenses, this 19th-century physics…

  • Mastering the Poisson Distribution: Intuition and Foundations

    Mastering the Poisson Distribution: Intuition and Foundations You’ve probably used the normal distribution one or two times too many. We all have — It’s a true workhorse. But sometimes, we run into problems. For instance, when predicting or forecasting values, simulating data given a particular data-generating process, or when we try to visualise model output…

  • Are You Still Using LoRA to Fine-Tune Your LLM?

    Are You Still Using LoRA to Fine-Tune Your LLM? LoRA (Low Rank Adaptation – arxiv.org/abs/2106.09685) is a popular technique for fine-tuning Large Language Models (LLMs) on the cheap. But 2024 has seen an explosion of new parameter-efficient fine-tuning techniques, an alphabet soup of LoRA alternatives: SVF, SVFT, MiLoRA, PiSSA, LoRA-XS … And most are based…

  • Linear Regression in Time Series: Sources of Spurious Regression

    Linear Regression in Time Series: Sources of Spurious Regression 1. Introduction It’s pretty clear that most of our work will be automated by AI in the future. This will be possible because many researchers and professionals are working hard to make their work available online. These contributions not only help us understand fundamental concepts but…

  • Experiments Illustrated: How We Optimized Premium Listings on Our Nursing Job Board

    Experiments Illustrated: How We Optimized Premium Listings on Our Nursing Job Board Running experiments is a task that often falls to data scientists. If that’s you, congrats! It can be a rewarding and high-impact area of work, but also requires tools found outside the typical ML-heavy data science curriculum. Even with the best tools, only…

  • When You Just Can’t Decide on a Single Action

    When You Just Can’t Decide on a Single Action In Game Theory, the players typically have to make assumptions about the other players’ actions. What will the other player do? Will he use rock, paper or scissors? You never know, but in some cases, you might have an idea of the probability of some actions…

  • I Won’t Change Unless You Do

    I Won’t Change Unless You Do In Game Theory, how can players ever come to an end if there still might be a better option to decide for? Maybe one player still wants to change their decision. But if they do, maybe the other player wants to change too. How can they ever hope to…

  • Reinforcement Learning with PDEs

    Reinforcement Learning with PDEs Previously we discussed applying reinforcement learning to Ordinary Differential Equations (ODEs) by integrating ODEs within gymnasium. ODEs are a powerful tool that can describe a wide range of systems but are limited to a single variable. Partial Differential Equations (PDEs) are differential equations involving derivatives of multiple variables that can cover…

  • Honestly Uncertain

    Honestly Uncertain Ethical issues aside, should you be honest when asked how certain you are about some belief? Of course, it depends. In this blog post, you’ll learn on what. Different ways of evaluating probabilistic predictions come with dramatically different degrees of “optimal honesty”. Perhaps surprisingly, the linear function that assigns +1 to true and fully…

  • 4-Dimensional Data Visualization: Time in Bubble Charts

    4-Dimensional Data Visualization: Time in Bubble Charts Bubble Charts elegantly compress large amounts of information into a single visualization, with bubble size adding a third dimension. However, comparing “before” and “after” states is often crucial. To address this, we propose adding a transition between these states, creating an intuitive user experience. Since we couldn’t find…

  • Introduction to Minimum Cost Flow Optimization in Python

    Introduction to Minimum Cost Flow Optimization in Python Minimum cost flow optimization minimizes the cost of moving flow through a network of nodes and edges. Nodes include sources (supply) and sinks (demand), with different costs and capacity limits. The aim is to find the least costly way to move volume from sources to sinks while…

  • How To Learn Math for Machine Learning, Fast

    How To Learn Math for Machine Learning, Fast Even with zero math background Photo by Antoine Dautry on Unsplash Do you want to become a Data Scientist or machine learning engineer, but you feel intimidated by all the math involved? I get it. I’ve been there. I dropped out of High School after 10th grade, so I…

  • Bayes’ Theorem: Understanding business outcomes with evidence

    Bayes’ Theorem: Understanding business outcomes with evidence A practical introduction to Bayes’ Theorem: Probability for Data Science Series (2) Continue reading on Towards Data Science » Sunghyun Ahn Go to original source

  • Bird’s-Eye View of Linear Algebra: Left, Right Inverse => Injective, Surjective Maps

    Bird’s-Eye View of Linear Algebra: Left, Right Inverse => Injective, Surjective Maps If matrix multiplication isn’t commutative, then why don’t we have left and right inverses? Continue reading on Towards Data Science » Rohit Pandey Go to original source

  • The Intuition behind Concordance Index — Survival Analysis

    The Intuition behind Concordance Index — Survival Analysis The Intuition behind Concordance Index — Survival Analysis Ranking accuracy versus absolute accuracy Taken by the author and her Border Collie. “Be thankful for what you have. Be fearless for what you want” How long would you keep your Gym membership before you decide to cancel it? or Netflix if you are a series…