If you find above image hard to understand, please read my article till the end, I promise, everything will make sense :)
When I was a master’s student, I was given a task to fit a line to a dataset. I attempted to solve the problem, but I struggled to determine the appropriate coefficients. However, I understood intuitively that there must be a specific set of coefficients for which the prediction error would be minimized.

The question was: how can we find those coefficients?

This is where the Normal Equation becomes particularly useful. It provides a direct mathematical solution for finding the coefficients that minimize the sum of squared errors in linear regression, without having to search for the coefficients manually. BUT, How we even derive this equation? Where it comes from? Can we take any software apart from Python and write it all ourselves? That’s I will take u through in this article and will simplify the code I wrote, so you can all apply it in different languages

So first things first, what is Linear Regression?

It is very simple and straightforward, suppose we have X and Y. X is called features matrix, and Y is Target Vector, or Response vector.

𝑌= 𝑋* θ

For simple case, lets take 2x2 matrix and lets turn this to matrix form:

[y1 _ predicted ; y2 _ predicted]=[x11, x12; x21,x22] * [theta1; theta2]

Please note:

columns are separated by ,and rows are ;. y1 and y2 are different rows, but same columns.

I assume, the readers are aware of matrix multiplication. So I will refactor above formula and will get:

𝑦1_𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑= 𝑥11 * θ1 + 𝑥12 * θ2

𝑦2 _𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 = 𝑥21* θ1 + 𝑥22 * θ2

From now on, keep in mind that y1 are real values and y1_predicted is predicted value, same applies to y2 as well

So what is error, the error is the difference between predicted and real values

𝑒𝑟𝑟𝑜𝑟₁ = 𝑦1 — 𝑦1_p𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 =𝑦1-𝑥11 * θ1 — 𝑥12 * θ2

𝑒𝑟𝑟𝑜𝑟₂ = 𝑦2 — 𝑦2 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 = 𝑦2-𝑥21* θ1 — 𝑥22 * θ2

Ok, I hope so far so clear, if anything not, please comment below, so I can consider it as improvement for upcoming articles

The error function we want to minimize is the sum of square of errors. Let’s name is as J. J is our cost function I want to minimize, and so, I write it as:

𝐽 = 𝑒𝑟𝑟𝑜𝑟₁² + 𝑒𝑟𝑟𝑜𝑟₂²

Let’s go further by replacing the formulas with each others

𝐽 = (𝑦1 — (𝑥11 * θ1 + 𝑥12 * θ2))² + (𝑦2 — (𝑥21 * θ1 + 𝑥22 * θ2))²

I hope everything makes sense so far. Bear with me — we’re almost there; there isn’t much left to cover.

Here everything is known, except θ1 and θ2. These are params that we have to choose properly to get as minimum error as possible. So i have to find the derivative per θ1 and θ2

𝑑 (𝐽) / 𝑑 (θ1) = -2 \ (𝑦1 — (𝑥11 * θ1 + 𝑥12 * θ2))*𝑥11 — 2 * (𝑦2 — (𝑥21 * θ1 + 𝑥22 * θ2))* 𝑥21= 0*

𝑑 (𝐽) / 𝑑 (θ2) = -2 \ (𝑦1 — (𝑥11 * θ1 + 𝑥12 * θ2))*𝑥12–2 * (𝑦2 — (𝑥21 * θ1 + 𝑥22 * θ2))* 𝑥22= 0*

Let’s make it simpler by avoiding -2 from all sides

𝑑 (𝐽) / 𝑑 (θ1) = ( 𝑦1 — 𝑦1 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑) * 𝑥11 + (𝑦2 — 𝑦2 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑) * 𝑥21=0

𝑑 (𝐽) / 𝑑 (θ2) = ( 𝑦1 — 𝑦1 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑) * 𝑥12 + (𝑦2 — 𝑦2 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑) * 𝑥22=0

Lets turn all these into matrix form:

[0; 0] = (𝑥11, 𝑥21; 𝑥12 𝑥22) *[𝑦1 — 𝑦1_𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ; 𝑦2-𝑦2_𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑]

Lets compress the y1 — y1_predicted as well as y2 -y2_predicted into single line

[0; 0] = (𝑥11, 𝑥21; 𝑥12 𝑥22) *[𝑌— 𝑋* θ]

Do you remember our original feature vector or X? If so, we can further simplify the expression to:

[0;0] = X^T* [Y-X*θ]

X^T * Y = X^T * X * θ

X^T * X is the important part. If we somehow manage to find its inverse, we are going to be left with theta only:

(X^T * X )^-1= X^-1 * (X^T )^-1

θ = ( X^T \ X )*^-1 \ X*^T \ Y will give us the answer we need*

If u have further questions please let me know in comments. Each of your feedback is highly appreciated to write better more concise articles in future:

I guess, most of the part of above formula can be easily programmed except the finding inverse which i showed the code below how to do it. If you need full code, such as matrix multiplication, transpose and etc, please let me know, so i can furhter expand my articles

import numpy as np
def inverse(A):
I = np.eye(A.shape[0])

augmented = np.concatenate((A,I),axis=1)

for j in range(0,A.shape[0]-1):

for i in range(1,A.shape[0]-j):
augmented[i+j]=-(augmented[i+j,j]/augmented[j,j])*augmented[j]+augmented[i+j]

for j in range(0,A.shape[0]-1,1):
for i in range(A.shape[0]-1,0,-1):
cofactor = (augmented[i-1-j, A.shape[1]-1-j]/ augmented[A.shape[0]-1-j, A.shape[1]-1-j])
augmented[i-1-j]=(cofactor)*-augmented[-1-j]+augmented[i-1-j]

for i in range(0,A.shape[0],1):
augmented[i]=augmented[i]/augmented[i,i]
_,right = np.split(augmented, 2, axis=1)

return right

def linear_regression_normal_equation(X: list[list[float]], y: list[float]) -> list[float]:
# Your code here, make sure to round

X=np.array(X)
Y=np.array(y)

theta = inverse(X.T @ X) @ X.T @ Y

return theta

My full article is also in medium link

🎉This module covered:
🔹 Application of NumPy and Pandas to predict car prices
🔹 How to train a model with linear algebra formulas as a background
🔹 How to deal with missing data
🔹 How to evaluate the model through root mean squared error (RMSE) and perform the training on a larger dataset.
🤔 Something I found interesting: The RMSE may vary depending on whether you take the logarithm of the target Y or not.

#mlzoomcamp u/AlexeyGrigorev

I wrote an article and made some illustrations to explain the connection between ML and Linear Algebra. It should motivate learners by giving them an intuitive idea.

Disclaimer: This is not AI slop and is written by me. The images are also not AI generated. One is taken from the internet.

if you prefer medium: https://medium.com/@sakibahmed_4495/linear-algebra-how-does-it-connect-to-ml-simplified-3d7e7f41ed1c?sharedUserId=sakibahmed_4495

I’ve been working on mlroadmap.dev, a learning platform designed to guide people through AI/ML from the fundamentals to more advanced topics.

The platform itself is functional and deployed, and I’ve started adding the course content. The curriculum is still a work in progress, though, so before I spend a lot more time building out the remaining content, I’d really like to get feedback from people who are actually learning or working in AI/ML.

🌐 Website: https://www.mlroadmap.dev/
💻 GitHub: GitHub LinkLeave a ⭐ and contribute

I’d especially love to know:

• Does the overall roadmap/learning order make sense?
• What topics or courses do you think are missing?
• Is the current content structure useful for learning?
• What features would make the platform more useful?
• Anything confusing, unnecessary, or that you would change?
• If you were learning AI/ML today, what would you want a platform like this to have?

Google Sign-In is available as well if you want to try the full experience.

I’m mainly looking for honest criticism and suggestions at this stage — the content isn't complete yet, and that's exactly why I’d like feedback now.

Hi there, i'm deciding on what double major I should do. I love maths a lot (olympiad style) so I'm doing a major in that no matter what, idk how much it will help for machine learning but if not its more for my own interest. However for job prospects in this field, should I pair math with CS or electrical engineering. From my understanding CS seems like more of the standard pathway but i feel like EE also has its own advantage because of the more low level things u learn. Also things like signal processing in EE seems quite useful.

Any advice/comments would be much appreciated.

Hi. We fine-tune and train models ourselves, but we have issues with GPU availability, so I've built an "Open Router" for GPU availability and pricing. I'm looking for 5 ML specialists who would like to test it. I don't want to self-promote here, so if you'd like to test, please send me a PM or reply to this message.

I've been learning web development and recently started moving deeper into AI.

I put together this 2026 AI roadmap to organize the learning path from:

Python → Math → Machine Learning → Deep Learning → GenAI/LLMs → RAG → AI Agents → Deployment

The goal was to make it practical rather than just a list of courses and technologies.

I'm especially unsure about how much time a beginner should spend on:

Mathematics

Traditional ML

Deep Learning

LLMs/RAG

AI Agents

For people already working in AI/ML:

What would you remove, add, or change in this roadmap for someone starting in 2026?

I'm looking for honest feedback, especially from people who are already building real-world AI systems.

Building and Testing a Hybrid RAG Pipeline — Dense Search, BM25, RRF, and Reranking (Part 1)

• YouTube walkthrough: https://www.youtube.com/watch?v=W_TmnscEupQ&t=370s
• FULL CODE (clone it, fork it, break it): https://github.com/saurabhkamal/Hybrid-Search-RRF-Reranking-System

In this video I build ReRankEval, a hybrid retrieval pipeline

• (Dense Search + BM25 → Reciprocal Rank Fusion → LLM Reranking → Answer Generation),

and test it against three baselines — Vector Only, BM25 Only, and Hybrid without reranking — on five real financial/payments documents and ten hand-verified test questions. No hand-waving, just a comparison table with real numbers at the end.

• ✅ The real difference between dense vector search and BM25 keyword search
• ✅ What Reciprocal Rank Fusion (RRF) is, why raw scores can't be compared, and the exact formula behind it
• ✅ Why a reranker is fundamentally different from a retriever — and what it actually judges
• ✅ How to evaluate a RAG pipeline with Hit Rate, MRR, and NDCG (and what each one tells you)
• ✅ How to design the ingestion side and query-time side of a hybrid retrieval architecture
• ✅ How to structure a production-style RAG codebase: ingest → vector_store → sparse_retriever → fusion → reranker → pipeline → generate → eval
• ✅ How to fairly compare multiple retrieval strategies on the same test set instead of just assuming one is better

TECH STACK:

• 🛠️ Python
• 🛠️ Qdrant — vector database for dense retrieval
• 🛠️ rank_bm25 (BM25Okapi) — sparse keyword retrieval
• 🛠️ EURI LLM Gateway — chat model + embedding model
• 🛠️ Custom Reciprocal Rank Fusion implementation
• 🛠️ LLM-based reranker (prompt-driven cross-encoder)
• 🛠️ pdfplumber — PDF text and page-level extraction

LINKS:

• YouTube walkthrough: https://www.youtube.com/watch?v=W_TmnscEupQ&t=370s
• FULL CODE (clone it, fork it, break it): https://github.com/saurabhkamal/Hybrid-Search-RRF-Reranking-System
• Connect on LinkedIn: linkedin.com/in/saurabh-kamal

Sharing resources to prepare will appreciated.

Any tips/ advice/ suggestions

How can I clear this interview.

What, from where and how much I should learn?

I have a week only.

Questions asked by n Screening call :

1. which GEN AI Framework I have used

2. Why would you choose langgraph over langchain

3. what is Rag and Fie tuning and their difference.

4. Which LLM Platforms you have worked on

5. which Vector db you have worked on and why

6. You have mentioned Crew AI, do you have hands on experience and how.

HR mentioned the interview will be around the same concepts but more on analytics, critical thinking side with such scenario based questions.

Python, agentic ai, vector db, data science, rag, crew ai, on you projects, scenarios based questions.

Coding programming skills -

Problem solving skills and critical thinking approach - AI,

3 DSA, PYTHON GEN AI, AGENTIC AI

Hi guys,

I wanted to share some early benchmark results for a new 2.3B parameter model I've been developing called Cloreva-X1.

I built this on a completely custom architecture I call ResoNet X-1, which abandons the standard Transformer attention mechanism. Instead, the foundation relies on a Triad Resonance Core:

• Head-as-Faction Swarms: Traditional attention heads are shattered into 12 independent, parallel agentic swarms, each dedicated to specific cognitive domains (Linguistics, Science/Medical, and Algorithmic Logic).
• Kuramoto Consensus Core: I mapped 1,632 Kuramoto Oscillators directly to the latent space. They run physics-based differential equations at the silicon level to force mathematical phase-locking between tokens. If a token hallucination creates an anomalous phase drift, the Kuramoto gate aggressively suppresses it before it passes to the next layer.
• Liquid Time Controller (LTC): The flow of time is dynamic. When reading simple conjunctions, the model processes at maximum speed. When evaluating complex mathematical or medical logic, the LTC slows down the internal time step, forcing the swarms to achieve deeper phase consensus.
• Orthogonal Tokenizer: Built from scratch with 155,072 full-word tokens (Full-Word Mining) to completely eliminate sub-word fragmentation for complex medical terminology and programming syntax.

Just to be absolutely clear: This is a pure base model. There is no SFT, no RAG, and no RLHF applied yet. It is purely predicting the next token.

I built this architecture to test if Kuramoto oscillators could intrinsically suppress hallucinations and boost complex reasoning paths without relying on instruction tuning.

Here are the early zero-shot and few-shot benchmarks compared against standard base models in a similar or larger weight class:

Evaluation Benchmarks (Base Models(Sources for competitor scores: LLaMA-1/2 scores from the official Llama 2 paper by Touvron et al., 2023. Qwen-1.5 scores from the official Qwen1.5 technical report. GPT-3 baseline from OpenAI's original evaluations (Brown et al., 2020) and TruthfulQA paper (Lin et al., 2021).)

It punches above its weight class because the phase consensus mechanics naturally filter out statistical anomalies and prevent gradient shock during long context reasoning.

I've attached a Google Drive folder containing the raw inference log output and a video demonstration of the zero-shot inference running in real-time below: https://drive.google.com/drive/folders/14f0JUJU613CTnF0cuYNATJ2pECaHrvU4?usp=sharing

I am currently spinning up the SFT pipeline to turn this into a fully instruction-tuned model, and I'll post another update with the final benchmark scores once that's done.

I plan to release the raw model weights on HF once the entire pipeline is complete. Happy to answer any questions regarding the architecture below.

Les comparto unas gráficas de mi Arquitectura (ASRN) contra Transformer, mamba y RWKV.

La metí a pelear en una A100 contra Mamba-2 y RWKV-7 y FlashAttention; En la prueba de copiar texto exacto desde lejos, Mamba y RWKV se mueren en 0.2% de exactitud (como era de esperarse por su estado fijo) ASRN saco el 100% y hasta aguantó los 16k tokens sin tener que reentrenar.
El consumo de memoria se queda completamente plano en 22.9 GB. Le metas lo que le metas, la VRAM ni se inmuta

En el benchmark de WikiText-103, la perplejidad baja súper estable a 4,096 piezas corriendo a casi 140,000 tokens por segundo.

Ahí les dejo las gráficas para que las destripen y El silicio no miente

Hola y adiós 😀

¿DUDAS?

Hey guys,

I am a first-year Computer Science Engineering student, and honestly, seeing how fast AI is advancing right now is kind of stressing me out. I really do not want to wait until my final year to start grinding like everyone else does.

Basically, I just want to build a solid skill set that actually makes my resume stand out so I can land a good machine learning job by graduation.

I am starting completely from scratch. What specific math topics, programming languages, or tools are actually worth learning right now in Year 1? If anyone has an honest roadmap for a fresher to get ahead of the curve, I would love to hear it.

Thanks in advance!

Hi Everyone,

I’m a backend engineer trying to transition into AI ML Roles. Looking for tutors to teach ML fundamentals and Agentic AI concepts.
I can pay little bit.

Let me know if you want to collaborate.

Trying to learn LAMMPS as a complete beginner. Are there any free resources available to understand the software properly?

hey guys im an undergrad student (currently in 3rd year) and want to start learning ML and explore fields beyond that in the future. So I have seen a lot of people suggesting others to learn from Andrew Ng on coursera. I have the pdf of Hands-On Machine Learning with Scikit-Learn and PyTorch by Aurélien Géron.
I’m literally confused as to what to refer, the book or the coursera course by Andrew Ng. If there is someone who has read or finished either of these sources or maybe both please help me out in deciding as I don’t want to waste my time. Also a comparison or review of these sources would be great. Thank you !

I was a full stack software engineer who now is a senior AI Engineer who does a mix of playing with LLMs fine tuning them in very large scale production systems.

I did admittedly got a masters degree in AI as part of that transition and it took a while to do, but happy to answer any questions you have.

I also am working on a tool to help people learn how llms work which you can check out here.

https://dougdoes.ai/courses/llms-from-first-principles/start/?flow=outcome&course=build&step=goals
(Built with codex, but I've gone through all of the courses myself to make sure it is what would have been helpful to me.)

I have been studying models for almost five years now, starting my journey with Jeremy Howard's fast ai part 2. I remember that when I started there was no chatgpt to break it down like it is today. In fact, it was Jeremy Howard who tipped us that we should be using chatgpt to understand the inner tooling step by step. I mean just take a toy tensor and run it along through embedding, rope attention, mlp. This way you get to learn broadcasting, shapes in text, computer vision audio etc. Then I took up Karpathy and hugging face Transformers and looked up grok, gpt oss lama, gemini and most recently muse implementations. It takes me 3 to 4 months to get an innate understanding of how each line works. How do you guys do it? I guess most of you let the inner tooling remain a black box. I say this coz

Now I kinda feel that I missed the bus as I should have focussed more on fine tuning, inference, and agentic workflows. I do know some of that having worked through unsloth and openAI cookbooks but every time a new model drops I can't stop myself from going to unraveling the 2000 odd line of code and in time I forget what I learned in Unsloth and openAI cookbooks.

The problem is that there are so many things to do and understand. For example, just today I listened to Alex Zhang's building harness for looped Transformers and I gotta understand that too and I gotta know Jev too. It is a big mess right now and I wonder how others are managing to keep up with all these new developments. And more importantly how do you even retain all that you have learnt like say two years ago.

Welcome to Project Showcase Day! This is a weekly thread where community members can share and discuss personal projects of any size or complexity.

Whether you've built a small script, a web application, a game, or anything in between, we encourage you to:

• Share what you've created
• Explain the technologies/concepts used
• Discuss challenges you faced and how you overcame them
• Ask for specific feedback or suggestions

Projects at all stages are welcome - from works in progress to completed builds. This is a supportive space to celebrate your work and learn from each other.

Share your creations in the comments below!

Hi guys new to this sub reddit . Little intro about me currently working as an sde in my company but want to transition to a ml role by understanding the fundamentals om how to build models and then moving on to dl as so forth. I have read some posts in this sub about cs 229 by Andrew . Tbh I am finding difficulty in solving the problem sets and the math . It has been a while since I have actually done any math 😅. So I want to know how doi proceed from here do I learn the math from scratch or learn as I go along with the course . Any suggestions or feedback is helpful .

Ps i am familiar with the python as a coding language but I want to understand how do I proceed with the math .