kipriyanovichJun 6
Alright, future engineers!
**Linear Independence:** Vectors are LI if no vector can be expressed as a linear combination of the others.
Ex: If `c1v1 + c2v2 = 0` only means `c1=c2=0`.
Pro-Tip: Essential for understanding bases & ensuring unique solutions in systems!
#VectorSpace #LinearAlgebra #STEM #StudyNotes
kipriyanovichApr 10
Alright, future engineers!
A **Basis** for a vector space is a minimal set of linearly independent vectors that can generate any other vector in that space.
Ex: `{[1,0], [0,1]}` is a basis for R^2.
Pro-Tip: Think of them as the fundamental 'building blocks' or coordinate axes for your space!
#LinearAlgebra #VectorSpace #STEM #StudyNotes
N-gated Hacker NewsApr 9
๐Ÿšจ BREAKING: #Nerds invent yet another way to squint at squiggles and call it progress! ๐Ÿš€ Apparently, mashing symbols together in "vector space" is the new black. ๐Ÿ™„ Remember, kids: when in doubt, slap on a fancy name and beg for donations. ๐Ÿ’ธ
https://charcuterie.elastiq.ch/ #Innovation #VectorSpace #TechHumor #Fundraising #HackerNews #ngated
Charcuterie

A visual explorer for Unicode. Browse characters, discover related glyphs, and explore scripts, symbols, and shapes across the standard.

Charcuterie
kipriyanovichMar 30

Alright, future engineers!

Vectors are linearly independent if no vector can be expressed as a linear combination of the others. Ex: `c1v1 + c2v2 = 0` only if `c1=c2=0`. Pro-Tip: If you can solve for any vector in terms of others, they're *dependent*! Crucial for bases.

#LinearAlgebra #VectorSpace #STEM #StudyNotes

HaraldJun 5, 2024

"You can use the metric to raise an index" is what I found in one or the other physics text book. Surely I missed some context each time, so I couldn't help to swear: ๐Ÿคข , I like the indexes where they are, what is the point of raising or lowering them.

I found the answer in Gravitation, by Misner, Thorne, Wheeler but felt the need to write it down in a way I understand it best: https://miamao.de/blog/2024-06/04.Raising_or_Lowering_an_Index.html

#physics #tensor #metric #vectorspace #dotproduct #raisinganindex #mtw

Haralds Blog โ€” Raising or Lowering an Index

Victoria Stuart ๐Ÿ‡จ๐Ÿ‡ฆ ๐Ÿณ๏ธโ€โšง๏ธJun 28, 2023

The Double Helix inside the NLP Transformer
https://arxiv.org/abs/2306.13817

We introduce a framework for analyzing types of information in a NLP Transformer. We distinguish four layers of information: positional, syntactic, semantic, and contextual.

We show that the distilled positional components of the embedding vectors follow the path of a helix, both on the encoder side & on the decoder side.
...

#NLP #MachineLearning #transformers #DependencyParsing #PartsOfSpeech #VectorSpace #informatics

The Double Helix inside the NLP Transformer

We introduce a framework for analyzing various types of information in an NLP Transformer. In this approach, we distinguish four layers of information: positional, syntactic, semantic, and contextual. We also argue that the common practice of adding positional information to semantic embedding is sub-optimal and propose instead a Linear-and-Add approach. Our analysis reveals an autogenetic separation of positional information through the deep layers. We show that the distilled positional components of the embedding vectors follow the path of a helix, both on the encoder side and on the decoder side. We additionally show that on the encoder side, the conceptual dimensions generate Part-of-Speech (PoS) clusters. On the decoder side, we show that a di-gram approach helps to reveal the PoS clusters of the next token. Our approach paves a way to elucidate the processing of information through the deep layers of an NLP Transformer.

arXiv.org
Show threadVictoria Stuart ๐Ÿ‡จ๐Ÿ‡ฆ ๐Ÿณ๏ธโ€โšง๏ธJun 15, 2023
#LLM #LargeLanguageModels #OpenAI #ChatGPT #tensors #NLP #NaturalLanguageProcessing #LargeLanguageModels #ML #MachineLearning #transformers #SelfSupervisedLearning #AugmentedModels #APIs #PromptEngineering #VectorSpace #VSM #ContextualLanguageModels #persagen #GenerativePretrainedTransformer #GPT #PretrainedModels #FineTuning #EmergentProperties #EmergingTechnologies
Show threadMarek GluzaMay 15, 2023

Juan-Pablo Ortega form #NTU recommended a GTM by Serre. I was aware of its existence but somehow assumed that it would be harder. Maybe I'll go for that instead?

Anyway, I like in Isaacs book that it exposes what aspects of linear #vectorspace make #GroupTheory tick. Also it was really fun reading Schur-Frobenius theorem yesterday, there the #module framework came to shine. I appreciate that.

What's more trouble for me is that the notation is sometimes not introduces certain things or I'm used to \(f(x)\) as opposed to \(xf\) ๐Ÿคทโ€โ™‚๏ธ. Im getting used to inline definitions made in passing but suddenly later coming into play.

Marek GluzaMay 14, 2023

I'm trying to educate myself in graduate level #mathematics and picked up this book by Isaacs on #GroupTheory characters. I managed to just so survive the first chapter which is a dense write up on #vectorspace #module theory. I'm able to read on, chapter 2 and 3 were still possible, but are there better entries into representations and characters?

Appreciate advice on:
1. easier but more specific exposition of central results?
2. advantages of developing everything in the A-module framework?
3. which chapters are key for subsequent developments?
4. which theorems are the ultimate highlights and which are 'just' key to the particular question at hand?
5. What's particularly worthwhile in the presentation approach taken in this book?

Show threaddr ๐Ÿ› ๏ธ๐Ÿ›ฐ๏ธ๐Ÿ“ก๐ŸŽงMar 22, 2023
@remi Using this for multiple tags and users would turn #Mastodon into a #vectorspace (linear combination of different feeds) where you can dial in your exact position of interest. "I need to #mastoart a little to the left and #math a little more towards to Z axis"