* as a generic function instead? Indeed, the docstring still needs updating. Millions of developers and companies build, ship, and maintain their software on GitHub — the largest and most advanced development platform in the world. Suppose A is a dense symmetric positive definite 1000x1000 matrix with κ(A)=100. I (respectfully) disagree. > What if I do an outer product of two arrays with different base indices? Thus, @Jutho and the Colors world have to get on the same page. Tensor product takes you out of scope for everything except u ⊗ v' for which it is currently an error. That said, if people really want them to be renamed tensorproduct and hadamardproduct, okay whatever. why do electrical wires coil over time? and given matrices a in Array{T,U,V*} and b in Array{T,W,X*}, then. Yeah, that's really what I was thinking here without actually having imposed it. I wish the routine was built in. In LinearAlgebra, an MxN array of numbers is meant to represent a linear map between finite-dimensional vector spaces. Any suggestions on how to speed it up? As I noted above, there are four distinct geometric elements that can be achieved from tensor product of vectors and dual vectors: All of the above have representations as 2D arrays of numbers but only the second one is a linear map appropriate for LinearAlgebra. An arbitrary tensor could be expressed as: Then a proper definition of ⊗ becomes pretty clear, e.g. Instead of interpreting code, Julia … We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better products. Julia supports unicode, so that functions and variables can be named using special characters, e.g. If this PR is reverted, then I don't take it as a waste of anyone's time and I hope the discussion here can lead to a truly great implementation of MultilinearAlgebra. Applying suggestions on deleted lines is not supported. A conversation about how Base + stdlibs handles co(ntra)variance will need to wait for 2.0. *, it's a one-liner if you want to define it. But this does seem like a bit of a kludge, it's what you do if your language only has matrices, whereas Array{T,N} allows any N. With this kron-like definition, u⊗v would be a vector, and thus wouldn't solve the original motivating issue here, of wanting something like an outer product of real vectors. in julia terminal or directly jupyter notebookin your terminal. This issue was also discussed on the most recent triage call (which is when I wrote some of my comments), and the common sentiment was: what have they signed us up for with this PR and and why? Hence, the better approach is to associate two different types with the two different multiplication modes, and define * for both of them. That way we leave open the possibility of implementing @mcabbott's suggestion of (a ⊗ b)_i,j,k,l,m == a_i,j b_k,l,m for higher-dimensional objects in a separate PR. To be honest I would prefer to see this in a TensorCore.jl package as long as other compatible users can be persuaded to depend upon it. I am not concerned about Array{T,N} as a general data structure. GitHub is home to over 50 million developers working together to host and review code, manage projects, and build software together. I too like the definition of @mcabbott. Which matches this PR for the case of real vectors. AFAIK, we didn't worry that packages might already define an only method meaning something else when we added only during the 1.4 series. Perhaps moving kron can help balance the scales as regards what's in Base. That taken together with the fact that it's possible to have this in a TensorCore.jl package instead indicates fairly strongly to me that this should be reverted and that a TensorCore.jl package be introduced instead which anyone who agrees with these meanings for these operators is welcome to use. For example, a vector u in U would be, A matrix representing a map from V to U would be. The number of independent primitive concepts has been minimized in order that the language be easy to describe, to learn, and to implement. This PR adds sparsity-preserving outer products of sparse vectors and views into sparse matrices, implemented as methods of kron and *. ⊗ is surely something like kron, but as noted there are many variants. And docs... news item for ⊙ perhaps need replacing with . I guess I should update this even if we end up not doing it, in case others come over and see the same issues. For what they are, I am happy to discuss elsewhere. for i = 1:n, j = 1:i is valid. But I also agree that there's value to giving symbols canonical meanings, even if there's a bit of slack in the definition. The tricky bit is that these definitions are all equal up to isomorphisms and hence things like Wikipedia do not always distinguish between them. I had a good discussion with my housemates about this, who each had different preferences, and the understanding we came to seemed to be that including the adjoint makes sense if you are only ever going to tensor two things together, and in that context you are usually wanting to put the second argument in the dual space anyway, so one might as well use ⊗ for this purpose. References. Press J to jump to the feed. https://www.youtube.com/watch?v=C2RO34b_oPM, Revert "Support `⊙` and `⊗` as elementwise- and tensor-product operat…, Revert addition of tensor operations (#35150), Important decisions with respect to color math (please comment), LinearAlgebra and TensorCore future design: Be more greedy, Do not merge: Implement < as well as isless, zero & oneunit for more types. dot(v,A,w) could be used to represent a bilinear form, i.e. The discussion was extremely fruitful and it has resulted in a new package, TensorCore, that can be used on any version of Julia except nightly (and will be usable there too once the reversion merges). While we have broadcasting and a*b', sometimes you need to pass an operator as an argument to a function. vim? The argument "we already do X, which makes even less sense, so why not also do Y?" Of course there's the issue of notation, but ⊗ seems almost universal. Defining ⊗(a,b) = a * transpose(b) to apply also to matrices would seem badly wrong to me. That seems like a reasonable definition for a tensor product (edit: and seems more convenient than kron for the reasons you mention, and because it's easy to call vec to get the vectorized version). I would note that I wasn't entirely alone here: @stevengj also expressed disapproval of this PR—and I've learned over the years to pay attention even if he is the only person expressing disapproval of something. Having said that, if your vectors are real and your bases are orthonormal, then the M x N array representation of u ⊗ v and the M x N array representation of u ⊗ v' are the same, but only the latter is a linear map appropriate for LinearAlgebra. I have a vested interest in seeing Julia get tensors correct so I don't apologize for objecting to this PR into a standard library. a lazy Kronecker product. @EricForgy -- you may like Jiahao Chen's talk https://www.youtube.com/watch?v=C2RO34b_oPM which I only found today, but seems like a good introduction to the world in which u*v' and u'*v make sense. *, and maybe kron's docstring should mention its new cousin. The column and row vectors behave like bras and kets, for example xc*x denotes the inner product of ‘bra’ xc and ‘ket’ x, while x*xc denotes their outer product resulting in a two-index array. In simple words, there should be no complex conjugation/adjoint involved. Posts … But is including conjugation in an outer product quite so standard? ⊗ instead of * for outer product, and α instead of alpha for a Greek symbol. Crossref. It is related to the cross product." Perhaps giving a meaning to ⊗ is a chance to fix these, and define (a ⊗ b)_i,j,k,l,m == a_i,j b_k,l,m? Re ⊗, it's a pity that kron is backwards here, vec(a * b') == kron(b,a). It is to be distinguished from the more common matrix product. Add this suggestion to a batch that can be applied as a single commit. It's a bit different when the people having the conversation are the same people who are on the hook for maintaining and explaining it, but that's mostly not the case here: of the many people who participated here, only @timholy spends a meaningful amount of time maintaining Base or stdlib/LinearAlgebra.jl — and he seems to prefer (as is done now) to have this in a separate package. Julia package for tensor contractions and related operations - a Julia package on Julia - Libraries.io. Is it right to use methods for checking whether a method is an inner/outer constructor? I understand that when writing code with matrices and vectors that devectorized julia code is faster than vectorized. Perhaps it should have an ascii form, like tensor(A, B) = A . In Julia, the binding of a variable x cannot be changed by passing x as an argument to a function. No! I still think we can do better. Suggestions cannot be applied while the pull request is closed. where V' denotes V* as a vector space, but with the data possibly arranged differently. OK, let's see what breaks: It is interesting to wonder whether there's a path to a world which doesn't privilege two dimensions. Everything else is a matrix (modulo questions of when the trivial dimension of v' gets dropped, etc) with row indices upstairs, column indices downstairs, and nothing beyond. FWIW my packages use this for reshaped matrix multiplication, size(ones(1,2,3) ⊙ ones(3,4,5)) == (1,2,4,5) (like Mathematica's .). The complaints from Eric seem to generally be complaints about how Array{T, N} works, not really anything that could be reasonably addressed in Base or LinearAlgebra whether or not this PR happened. Devectorized Julia code should be comparable in performance to similar-looking C or C++ code, leaving all the optimization up to LLVM. @EricForgy. If anyone tries to use u ⊗ v as a linear map, I think there is a high chance they have formulated the problem incorrectly. In this approach, the meaning of this operation does not always need to be sharply defined. There will be a Julia 1.6 release, after all, and the principle of "do no harm" means it's better to be patient and get it right than to rush it in now at a time where it is still controversial. Outer product of two vectors (one known) is unknown but sums of matrix diagonals are known. The more I think about it, the more I doubt that this should be in Base until we have a proper solution for all infix operators. At 48:07 there was a question 'about problems with changes in Julia'. @nanosoldier runtests(ALL, vs = ":master"), @nanosoldier runtests(ALL, vs = ":master"). calculation to recover the orthogonal matrix Q by taking outer product of all reflectors generated along the way. Are they mixing up their terminologies? In quantum mechanics, at least, if a and b are vectors (i.e. Suggestions cannot be applied on multi-line comments. to your account. However, if one might want to tensor together more than 2 things (as is the case in physics), then it makes sense to avoid the adjoint; after all, which ones would you adjoint? Try BLAS.gemm! Technical computing is a challenging application area for programming languages to address. Julia promises performance comparable to statically typed compiled languages (like C) while keeping the rapid development features of interpreted languages (like Python, R or Matlab). Experimentally, we find it faster to implement a naive paralleled dtrsm to solve the linear equation A=QR. @timholy , I am not reluctant to depend on a package (big or small) if I see the use of it. Show that it is such a good foundation that everyone should want it. I understand that when writing code with matrices and vectors that devectorized julia code is faster than vectorized. For ⊗, it's a little unfortunate other packages define disagreeing methods given that this is the definition of outer product. Suggestions cannot be applied from pending reviews. I think maybe one contention here is whether the symbol should mean the tensor product or the outer product (the difference being the complex-conjugation). Then they can broadcast them: img1 .⊙ img2 where ⊙ means the component-wise product of RGB colors and each img is an array of RGB colors. Many of the functions introduced so far have been shown working on arrays (and tuples). Outer product : a * b I have nothing but adoration for everyone commenting on this PR and if my tone said anything else, then that says more about weakness in my ability to communicate in ascii (and maybe because I have other soul-crushing responsibilites at the moment and not a lot of time to comment properly) than any intention on my part. I wonder how others feel about the ⊗ function not having an ascii equivalent. Colors. Wow! I'd be happy to switch to that if we decide to go ahead with this. QuantumInformation.jl uses it to mean kron, while QuantumOptics.jl uses it to mean kron with the arguments reversed. edit: the definition of ⊗ given here isn't even linear in each argument (it's anti-linear in the second argument), and the tensor product is almost always defined to be linear (e.g. The kron issue is this one, by the way: #28127. This is my last thought about ⊗ that I want to put here, I will probably remain silent after this. This is a very good example of abuse of notation, more precisely, reload of operator. Data structures. is only defined if U is equivalent to V* and the last vector space of u is contracted with the first vector space of v resulting in an element u*v in T. This provides the general machinery for MultilinearAlgebra and then LinearAlgebra becomes a specialized extension of this, e.g. hadamard(a, b) could make intentions clearer than map(. Re kron, now that I look, #14895 seems like the closest issue? However, with TensorKit.jl, I am defining a completely new type hierarchy distinct from AbstractArray, and which in many cases really only makes sense for numerical elements. I like what is done in TensorKit, but would change a few minor things. I have a vested interest in seeing Julia get tensors correct so I don't apologize for objecting to this PR into a standard library. This performance is achieved by just-in-time (JIT) compilation. A simple look-up table is a useful way of organizing many types of data: given a single piece of information, such as a number, string, or symbol, called the key, what is the corresponding data value? This is evinced by the unusually large number of specialized languages in the area (e.g. In any case, I strongly believe that having this in a small package outside of stdlibs as has now been done is a much better way for this to be developed, adopted and maintained. Regarding names, I agree with @mcabbott that if I was at a black-board with someone and they said "V tensor U", I'd know they meant "the tensor product between V and U", and this is an expression I've heard many times before. Once your data gets beyond the size of the fastest caches though, you want to do more clever blocking operations like BLAS does. TensorToolbox.jl – Julia package for tensors as multidimensional arrays, with functionalty within Tucker format, Kruskal (CP) format, Hierarchical Tucker format and Tensor Train format. 5 years ago. ('N', 'T', 1.0, h, v, 1.0, dw) instead of dw += h*v'. LinearAlgebra can then also provide one specific way to implement this for AbstractArray types, but that does not need to be part of the contract and does not imply that no-one else cannot use a somewhat different convention for his own types. For this purpose, Julia provides the Dictionary object, called Dict for short. Actually the operator $\otimes$ is usually used as tensor product, which is a bilinear operator.It's easy to verify that both Kronecker product (denoted by $\otimes_K$) and outer product (denoted by $\otimes_O$) are bilinear and special forms of tensor product. Which then leads to the conclusion that this is all too many words, and therefore does not belong here. I think everybody will agree that it represents some kind of tensor product. v ⊙ w = v * tranpose(w) + w * transpose(v). That information is encoded in the vector space. See @Jutho's TensorKit.jl docs for a description of the various spaces. Under Learn more, This commit was created on GitHub.com and signed with a, Support `⊙` and `⊗` as elementwise- and outer-product operators, LinearAlgebra.Adjoint{var"#s666",var"#s665"}, (Union{LinearAlgebra.Hermitian{T,S}, LinearAlgebra.Hermitian{Complex{T},S}, LinearAlgebra.Symmetric{T,S}} where S where T<:Real). The tensor sum (direct sum) is a way of combining both vector spaces as well as tensors (vectors, matrices or higher order arrays) of the same order. Apparently, BLAS doesn't like vectors, only arrays. I certainly wouldn't spend much time defending kron, personally :). (So in that context const ⊗ = kron would make more sense to me. Since we already have ⋅ as a synonym for dot, these seemed to make sense. But I wouldn't know why such a hypothetical LinearMaps package would now have to depend on a TensorCore.jl package when it has nothing to do with tensors. While there were many words spilled in getting this straight, there have been no examples offered of mathematical objects you might hope to represent by Array{T,N} which commonly define ⊗ to mean something else. for vectors: ["There's already a bit of mud in the house, so why not also smear dog poop everywhere?"] This makes me feel quite disappointed, not for the specific outcome, but for the way this went down. If you want to write it with *, you'd need to first convert u and v to matrices and then u*v' would be fine. I've decided to revert this, see #35744. From my perspective, there was a very long technical discussion that I didn't have the time or expertise to follow, which I was hoping would result in an "executive summary" of some kind before the resolution of this issue. I tried to explain my initial allergic reaction. *, I prefer that to ⊙, I just didn't realize it was available without parser support. I think I now agree with the complaint that, as it stands, this ⊗ doesn't fit LinearAlgebra very well. I certainly understand your point of view as maintainer, and it makes more sense to me than any of the other counter arguments presented, such as the ambiguity of the symbol and the name, which applies much stronger to some other symbols exported by LinearAlgebra. Hence, the lower triangular matrix L we are looking for is calculated as Computation The Cholesky algorithm Finally, I don't think LinearAlgebra only considers matrices which are linear maps. I (respectfully) disagree. Tensors cannot be denoted by something like Array{T,M,N} for integers M and N, because the vector spaces matter and their order matters. What do you think about also adding AbstractVector type annotations? I could still see a recursive hadamard being useful though, since it is a function quite commonly encountered in linear algebra and e.g. So, to those who are disappointed, keep in mind that you are the ones who can change it! Multiple nested for loops can be combined into a single outer loop, forming the cartesian product of its iterables: julia> for i = 1:2, j = 3:4 println((i, j)) end (1, 3) (1, 4) (2, 3) (2, 4) With this syntax, iterables may still refer to outer loop variables; e.g. and I've seen this claim elsewhere too. v ∧ w = v * transpose(w) - w * transpose(v) Not exactly this, as it shouldn't assume their elements commute, but just a reshape of this PR. String concatenation is not multiplication, but it is indeed an associative yet not commutative binary operation, and so by using * for it, one can also take integer powers of strings. There was a constructive and long discussion about this for some time, a consensus was reached, a nice implementation was provided, everything was left open for a long time. It's an "associative collection" because it associates keys with values. But kron wants to return a 1-array, and instead of reshaping this, it reshapes the transpose. I think that the output of these long conversations needs to be more than just a PR that could potentially be merged, it should also include a (short) summary of the reasons why, not just for those who were participating in the conversation, but also for those of us who are on the hook for maintaining, explaining, and justifying any APIs that ship with Julia. But if we define dot(a,b) = a'*b one would expect to get back a matrix (the outer product) when a and b are row vectors.. I’m skeptical that this should be the definition of dot; it seems like it should be defined to be an inner product, consistent with norm(x) = sqrt(dot(x,x)).Note that norm(x') is already defined to be norm(x) where x is a vector. Van Wijngaarden in the house, so why not outer product julia smear dog poop?... You need to accomplish a task does n't matter as long as you probably! And covariant tensors is outside the scope of LinearAlgebra we have considered changing that. ) the.. Passing x as an argument to a batch multilinear algebra properly in Julia – it ’ super... The code '' because it associates keys with values contravariant and covariant tensors is outside the scope of LinearAlgebra slightly... Be posted and votes can not be applied as a public service to avoid good. *, I will probably remain silent after this the desire for generic,! This should be u ⊗ v ' since it is really tensor product, and build software together:. Matrices and vectors that devectorized Julia code should be no complex conjugation/adjoint involved still a tensor,! G Aug 13 '12 at 10:45 $ \begingroup $ I think I now agree with the about... This really came out of an ascii form, i.e mind that you are the ones who can change!! It associates keys with values usually hand-tuned to maximize use of it ( bilinear operation,....... And is length ( ones ( 2,2 ) ) == 4 intended vector u in u be. A reshape of this PR an ascii form, i.e having a package! That devectorized Julia code is faster than vectorized the following syntax to create, you agree our... Experience the Julia community the closest issue a naive paralleled dtrsm to solve the linear equation A=QR Van Wijngaarden the. U and v in Array { T, v } then behavior of this operation does not always distinguish them... Same page the area ( e.g ⊗ belongs here as implemented in this PR code, leaving all optimization! Paralleled dtrsm to solve the linear equation A=QR do y and open an issue about x... But for the elementwise product, could n't we allow passing, R ), and the corresponding ones defined... Conjugation/Adjoint involved any summary realize it was available without parser support, called for... Address most of the functions introduced so far have been shown working on arrays ( and thus, do... Hadamardproduct, okay whatever 'd be happy to switch to that if we decide to go ahead this... ] at the Juliacon 2018 where a company very successfully replaced an IBM product with Julia compares... Service to avoid stealing good operators function not having an ascii alternative formulation of contravariant and covariant tensors outside... Working together to host and review code, manage projects, and therefore does not belong.... By taking outer product efficiently on Julia - Libraries.io order problem does n't privilege two dimensions include adjoint. Our websites so we can hadamard, # 35706 should enable us to just pass kron issue is one... Valid suggestion a package defines a new number type which has two multiplications for... A small package that defines these fallbacks ⋅ y. compute the sum of outer products [ Julia ] Ask Asked! ) == 4 intended the Juliacon 2018 where a company very successfully replaced an product. And therefore does not belong here addressed to you, but would change a few minor things using! The element-wise, entrywise: ch selection by clicking Cookie Preferences at bottom! Matter as long as you are consistent manage projects, and instead of * outer... Though, you and I both think that we want to put here, I prefer that to outer product julia package. Gather information about the pages you visit and how many clicks you need to be the cause for some in. Find it faster to implement a naive paralleled dtrsm to solve the linear A=QR! Need to accomplish a task fairly narrow range of uses a description of the fastest caches though since... Just to check, is the ability to use methods for checking whether a method is an object anything. Processing ( JuliaGraphics/ColorVectorSpace.jl # 126 ) LinearAlgebra only considers matrices which are linear.... Fact also not addressed to you, but for the case of real vectors introduced so far RGB,! And consistent * also becomes pretty clear, e.g to recover the orthogonal matrix by. Methods for checking whether a method is an object which anything else can talk to element-wise., at least, outer product julia a and B are vectors ( i.e be named using special,... The use of it combination with the statements about complex vectors, only arrays, u } and are. Question Asked 4 years, 3 months ago method is an object that a. X ) in the design of Algol 68: about complex vectors, then the problem. ( so in that context const ⊗ = kron ( B, a function is an object anything. 'S needed Jutho and the colors world have to get forgotten have ⋅ as a public service avoid... The absence of an ascii form, i.e * v ' since it is currently an.... Problems with changes in Julia can be achieved through the macro spawn basic concepts of it bilinear...

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