This is an example of amortized complexity and Strassen's "asymptotic spectra" (nice monograph by Zuiddam & Wigderson: https://www.math.ias.edu/~avi/PUBLICATIONS/WigdersonZu_Final_Draft_Oct2023.pdf)

Strassen developed this to understand the #complexity of matrix multiplication and #tensors, but it turns out to also show up in a bunch of places:
- #Entropy
- #Quantum information
- Shannon capacity of graphs
- Communication complexity https://en.wikipedia.org/wiki/Communication_complexity#Information_Complexity
- Circuit complexity (Robere & Zuiddam https://eccc.weizmann.ac.il/report/2021/035)

#math #probability #ComputationalComplexity #TCS #InformationTheory

Ah, nothing screams "exciting" like a deep dive into #PyTorch internals! 🎉 Let's unravel the mysteries of #tensors, because who doesn't love a bedtime story about C codebases? 💤 Spoiler: it's as thrilling as watching paint dry, but with extra parentheses. 🤓
https://blog.ezyang.com/2019/05/pytorch-internals/ #CCodebase #DeepDive #TechHumor #ProgrammingInsights #HackerNews #ngated

hello, I have a couple questions about #physics more precisely #relativity

How do I actually use #metric #tensors? Do I just do a dot product over the elements with a vector? If a metric tensor has a variable inside, does that variable comes from the vector and I just replace it? After I replace the vector and do the equation, only then I do the dot product?

If I want to have the inverse of a metric tensor, do I just do a #matrix inversion?

#physicsquestion #pleaseboost #pleaseboostme

🌟 What are "Tensors," and why are they at the heart of machine learning and AI? 🤖
My first ever #DevBytes post, I break down how these multi-dimensional powerhouses shape the world of data representation, computation, and neural networks. Perfect for beginners and enthusiasts alike!
💻 Dive into it here: https://smsk.dev/?p=148

Let me know your thoughts—I’d love to hear from you! 🚀

#MachineLearning #DeepLearning #Tensors #AI #DataScience

🚀 **Introducing DevBytes**: Quick, fun dives into software & hardware engineering!
I’m launching a new blog series that breaks down software concepts with bite-sized insights and hands-on tips. From **tensors in LLM models** 🧠 to quick coding fixes, there’s something for everyone.
Get ready to explore theories, best practices, and more—delivered in minutes.⏳
👉 Check it out here: https://smsk.dev/2025/02/14/introducing-devbytes-quick-dives-into-software-engineering/

#DevBytes #SoftwareEngineering #CodingTips #Programming #HardwareEngineering #ai #LLM #Tensors #ShortFormLearning

'Guaranteed Nonconvex Factorization Approach for Tensor Train Recovery', by Zhen Qin, Michael B. Wakin, Zhihui Zhu.

http://jmlr.org/papers/v25/24-0029.html

#tensor #tensors #factorization

'A tensor factorization model of multilayer network interdependence', by Izabel Aguiar, Dane Taylor, Johan Ugander.

http://jmlr.org/papers/v25/23-0205.html

#tensors #tensor #multilayer

#Pytorch is really good supporting #tensors | Can run across multiple clusters and multiple platforms #allthingsopen #AllThingsOpen2024

#Tensors Everywhere in #Complexity

Me! At the colloquium.

With connections to geometric complexity theory, graph isomorphism, group isomorphism, #quantum entanglement, and post-quantum-secure cryptosystems.

Online Fri Oct 25 2024 at IU Bloomington CS: https://events.iu.edu/siceiub/event/1663944-tensors-everywhere-in-complexity

#math #ComputationalComplexity #CS

Good high level overview of what #tensors are
https://www.quantamagazine.org/the-geometric-tool-that-solved-einsteins-relativity-problem-20240812/

Tensors from scratch series #tensors #math #ai https://www.luisquintanilla.me/feed/tensors-from-scratch-blog-series?utm_medium=feed

Just stumbled onto this playlist showing how to visualize tensors, specifically F𝝁𝝂, the electromagnetic field tensor. Looks like the animation was made with manim but I bet it could be done in (Web) VPython. #ITeachPhysics #ElectromagneticField #Tensors

https://youtube.com/playlist?list=PL2aHrV9pFqNTEMuDFre16Wx2SwBCNiR7j&feature=shared

e.g. if I tell you I have a matrix M and under change of basis it transforms as A^t M A, then I know it's representing a bilinear map of the form V⊗V→F. etc.

Similarly, if I tell you what kind of multilinear "thing" a tensor T is representing, then that tells you how it transforms under change of basis, and vice versa. For 3-tensors, there are several natural possibilities (up to permuting indices):

U⊗V⊗W→F
U⊗U⊗V→F
U⊗U⊗U→F (trilinear map)
U⊗V→W (bilinear map)
U⊗U→V (bilinear map)
U⊗V→U (linear action of V on U)
U⊗U→U (algebra, not nec. associative)
U→V⊗W
U→U⊗V (coaction)
U→U⊗U (coalgebra, not nec. coassociative)
F→U⊗V⊗W
F→U⊗U⊗V
F→U⊗U⊗U

(4/4)

#tensors #matrix #algebra

How the matrix transforms is "equivalent data" to "what kind of multilinear thing the matrix represents."

(3/4)

#tensors #matrix #algebra

Now, how do we get the whole "a tensor is a thing that transforms like a tensor"? Well, let's start with matrices. How a matrix changes under change of basis *tells you what kind of multilinear thing the matrix is representing*, and the same is true of tensors. Examples:

If a matrix M represents a linear map L:V→W, then when we change basis in V by an invertible matrix A in GL(V), and change basis in W by an invertible B in GL(W), then M changes to B M A^{-1} (where I'm writing my inputs as column vectors on the right).

In contrast, if a matrix M represents a linear endomorphism L:V→V, then when we change basis in V by an invertible matrix A in GL(V), M becomes AMA^{-1}.

If a matrix M represents a bilinear map V⊗V→F (by (x,y)→x^t M y), then under change of basis A^{-1}, M becomes A^t M A.

(2/4)

#tensors #matrix #algebra

No one defines a #matrix as "a thing that transforms like a matrix". Why define tensors that way?

Array=numbers in a (possibly high-dim) grid
Matrix=array representation of a linear map* in a chosen basis
Tensor=array representation of a multilinear map in a chosen basis

(* or linear endomorphism, or bilinear function, but we'll get there.)

Vectors=1-tensors, but not all 1-index arrays are vectors
Matrices=2-tensors, but not all 2-ary arrays are matrices

Similarly, not all k-ary arrays are tensors. Some examples:

Christoffel symbols aren't a tensor because they aren't (multi)linear in all of their arguments.

Most "tensors" in #MachineLearning #AI aren't tensors b/c they aren't multilinear - they are *just* multi-dim arrays of numbers. To say an array is (or represents) a tensor is to endow it with additional multilinear structure, same as with arrays vs matrices vs linear structure.

(1/4)

#tensors #matrix #algebra

A Counter-Counterexample!!

Comon’s Conjecture is undead now.

https://epubs.siam.org/doi/full/10.1137/23M1623781

#tensors #formalizationFixesThis @MoritzFirsching

#Spinors can be viewed as "square roots" of #sections of #vector #bundles.

https://en.m.wikipedia.org/wiki/Spinor

#Spinors are mathematical entities somewhat like #tensors, that allow a more general treatment of the notion of #invariance under #rotation and #Lorentz boosts.

https://arxiv.org/abs/1312.3824

https://youtu.be/b7OIbMCIfs4

I wrote about how I’m building a generic tensor library for .NET, using SIMD and value-type delegates to optimize performance and reduce code duplication.

Check it out here: https://aalmada.github.io/A-generic-tensor-library-for-dotnet.html

#dotnet #csharp #simd #tensors

Nerding out on #eigenchris ' wonderful math videos. Just finished the Tensor for Beginners series, going on to Tensor Calculus. Hugely important, as #tensors are the gateway to the theory of #relativity among others. #math #physics
https://www.youtube.com/playlist?list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG