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# outer product julia

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). New comments cannot be posted and votes cannot be cast. Tim said: For ⊗, it's a little unfortunate other packages define disagreeing methods given that this is the definition of outer product. *, I prefer that to ⊙, I just didn't realize it was available without parser support. I'd be happy to switch to that if we decide to go ahead with this. 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. So in that context const ⊗ = kron would make more sense to me. * is not defined, though we have considered changing that.). @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. This really came out of an extensive discussion regarding color arithmetic in image processing (JuliaGraphics/ColorVectorSpace.jl#126). Not sure we should use up ⊙ to mean . In that case, yeah, dot should probably just be a twice covariant tensor, but then you'd end up seeing things like dot ⊗ A , In my opinion, * should have a definition consistent with multilinear algebra. I don't see it that way. Or perhaps this is indeed another PR. give u1,u2 in U, then. Show that it is such a good foundation that everyone should want it. EDIT: I had to call reshape to turn the N-dimensional vectors 'v' and 'h' into Nx1-dimensional arrays in order to use BLAS.gemm. * reshape(B, ntuple(_->1, ndims(A))..., size(B)...), plus some smarter methods for B::Adjoint etc. And docs... news item for ⊙ perhaps need replacing with . 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. What is the fastest way to compute the sum of outer products [Julia] Ask Question Asked 4 years, 3 months ago. 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. We’ll occasionally send you account related emails. Yeah, that's really what I was thinking here without actually having imposed it. I've decided to revert this, see #35744. I'd argue that it's really their problem if LinearAlgebra imposes the correct meaning. Dear @StefanKarpinski, this comment was in fact also not addressed to you, but let's just forget about it. More generally, if a in Array{T,U,W*,V*} and b in Array{T,U,W,V*}, then. You can tell from the shortage of tests I wasn't thinking beyond a fairly narrow range of uses . It would be great to have an issue (is there one already?) Thus, @Jutho and the Colors world have to get on the same page. If a package defines a new number type which has two multiplications (for whatever reason, e.g. * as a generic function instead? This suggestion has been applied or marked resolved. I haven't read all of this but the existence of this many words to be spilled by different people on the subject tells me what I need to know—this is too contentious to go in a stdlib. The problem I see with having a separate operator for this is that beginners use this instead of dotted function calls and aren't necessarily aware that this prevents loop fusion. He was, for example, the only person who complained about the soft scope change before 1.0, which turned out to be kind of a big kerfluffle. This addresses the desire for generic concepts, nicely expressed by @Jutho. This PR adds sparsity-preserving outer products of sparse vectors and views into sparse matrices, implemented as methods of kron and *. cobordisms or tangles or ..., any objects and morphisms in a tensor category), one aspect thar all these choices agree on is that in a linear category, of which the category Vect of vector spaces is the prime example, the tensor product is a bilinear operation, i.e. calculation to recover the orthogonal matrix Q by taking outer product of all reflectors generated along the way. Hence, revert away. If A ⊗ B returns a 4-tensor, that's not an object which anything else can talk to. Other people try to reconstruct several of the previous arguments for how this consensus was reached in a detailed (and thus quite long) manner. Hence, the better approach is to associate two different types with the two different multiplication modes, and define * for both of them. Experimentally, we find it faster to implement a naive paralleled dtrsm to solve the linear equation A=QR. Julia supports unicode, so that functions and variables can be named using special characters, e.g. 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. Learn more, We use analytics cookies to understand how you use our websites so we can make them better, e.g. vim? I love to see a Julian speaking category theory It's a beautiful package ❤️ The only issue I have is that {M,N} is not really sufficient since the order matters (but that is not a discussion for here ). But is including conjugation in an outer product quite so standard? Or roughly ⊗(A,B) = kron(B,A) could be in LinearAlgebra. Crossref. A conversation about how Base + stdlibs handles co(ntra)variance will need to wait for 2.0. @EricForgy. Re ⊗, it's a pity that kron is backwards here, vec(a * b') == kron(b,a). I take that as an argument not to have tensor in LinearAlgebra since the only way to do it is to not take dual spaces seriously. Applying suggestions on deleted lines is not supported. Devectorized Julia code should be comparable in performance to similar-looking C or C++ code, leaving all the optimization up to LLVM. I am trying to compute an outer product of two large vectors, and the operation is very slow. Press question mark to learn the rest of the keyboard shortcuts, [JULIA] How to calculate an outer product efficiently. Following the discussion on Arrays in Julia 0.5 #13157, I want to suggest including new operators for the tensor sum (or direct sum) operation and for the tensor product (outer product).. I am not concerned about Array{T,N} as a general data structure. couldn't we allow passing . I would definitely contest that this is the correct definition of ⊗, though I admit it depends on the field. for vectors: Runs as fast as I hoped. From a physics point of view, I'm not really sure if I agree that transpose is the most natural definition of an outer product. Perhaps (within how LinearAlgebra thinks) this should be more like *′(a,b) = a * b'? exact and approximate, fast and slow, two different definitions, ...), and that package decides to use * and ⊗ for those two operations, that's of very limited use, because e.g. It is actually possible to get this to work already: Although this is quite hacky and the proper way would probably be to do this during lowering, as proposed in #34156. If you think about automatic differentiation, which involves pullback / pushforward, for example, these things should matter (I think). I could happily change this to not take the conjugate. @EricForgy, I just wanted to point out that in TensorKit.jl, M and N do not actually denote the upper and lower / contravariant and covariant indices. TensorToolbox.jl – Julia package for tensors as multidimensional arrays, with functionalty within Tucker format, Kruskal (CP) format, Hierarchical Tucker format and Tensor Train format. you just concatenate the list of vector spaces in the parameter. Learn more. That said, if people really want them to be renamed tensorproduct and hadamardproduct, okay whatever. I think the vector method should avoid recursive transpose: I was concerned that the reshape method would be slow for Adjoint{T,Matrix} etc, but it seems to work fine. That is a twice-contravariant tensor. why do electrical wires coil over time? dot(v,A,w) could be used to represent a bilinear form, i.e. See @Jutho's TensorKit.jl docs for a description of the various spaces. 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. ⊗ is surely something like kron, but as noted there are many variants. And addresses the objection of not quite fitting into LinearAlgebra. Contains as special case the outer product where no indices are contracted and a new array is created with a number of dimensions that is the sum of the number of dimensions of the two original arrays. Basictypesandoperations 4. Following the discussion on Arrays in Julia 0.5 #13157, I want to suggest including new operators for the tensor sum (or direct sum) operation and for the tensor product (outer product).. I.e. * to e.g. 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. 1: G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Baltimore, MD, Johns Hopkins University Press, 1996, pg. Dear @EricForgy , that comment was certainly not addressed to you (and I should probably not have made it anyway, this was a spur of disappointment). To me, Julia is so successful at inter-package functionality and generic programming, because Base or standard libraries assigns a meaning to a sufficient number of basic functions and operators, and than a user/package can just define his own types to specify how he wants to implement this meaning for his specific use case, by relying on multiple dispatch. > What if I do an outer product of two arrays with different base indices? Successfully merging this pull request may close these issues. Once your data gets beyond the size of the fastest caches though, you want to do more clever blocking operations like BLAS does. At 48:07 there was a question 'about problems with changes in Julia'. It is fine. This is a very good example of abuse of notation, more precisely, reload of operator. Hence, I don't see any use case where this is combined with e.g. I agreed with the conjugate at the beginning, but generalizing to higher-dimensional tensors seems to be the stronger argument. Suggestions cannot be applied on multi-line comments. ⊗ instead of * for outer product, and α instead of alpha for a Greek symbol. *, A, B) are what's needed? and I've seen this claim elsewhere too. This gets merged, one person shouts out that this is not good without all that much of motivation, or arguments which apply much more strongly to existing functionality, and frankly in a rather condescending tone. I think that everyone would agree that, for real vectors, (a ⊗ b)_i,j == a_i b_j, which is a 2-array (like this PR). This makes me feel quite disappointed, not for the specific outcome, but for the way this went down. 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. Whereas ⊗ works for Base's Array{T,N}, which treats all N, and has no notion of that the second index of an Array{T,2} is co/contravariant. Data structures. Viewed 908 times 3. You'll get an offset array indexed by the product of the axes of the input: julia> A = OffsetArray(1:3, 0:2) OffsetArray(::UnitRange{Int64}, 0:2) with eltype Int64 with indices 0:2: 1 2 3 julia> B = 4:6 4:6 julia… Sign in v ⊗ w should be an element of V ⊗ W, whereas linear maps W->V are isomorphic to V ⊗ W^*. My actual goal is to avoid having to define * for RGB colors and instead force people to specify which sense of multiplication they want: ⋅, ⊙, or ⊗. I certainly agree that getting a robust formulation of contravariant and covariant tensors is outside the scope of LinearAlgebra. No! Concepts like strings, arrays, dictionaries, tuples, and sets all behave in a manner that is more or less similar to those in Python. As I keep saying, the order is backwards for interop with column-major operations like vec and reshape which is why we need a new function! Are they mixing up their terminologies? References. In this approach, the meaning of this operation does not always need to be sharply defined. Generally a little slower, since BLAS is usually hand-tuned to maximize use of SIMD and multithreading. While we have broadcasting and a*b', sometimes you need to pass an operator as an argument to a function. I think I now agree with the complaint that, as it stands, this ⊗ doesn't fit LinearAlgebra very well. 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. Experimentally, we find it faster to implement a naive paralleled dtrsm to solve the linear equation A=QR. Neither definition agrees with the use here: for vectors a and b, one has the identity a * b' == kron(a, b') == kron(b', a), so they both agree with each other in this scenario, and disagree with this PR by an adjoint. Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results. (The implementation is careful not to take a view of LinearAlgebra's slightly odd Adjoint types.) I do think (aside from the possible exception of dot), LinearAlgebra should be constrained to vector spaces, dual vector spaces and linear maps between them. The kron issue is this one, by the way: #28127. Hence, the lower triangular matrix L we are looking for is calculated as Computation The Cholesky algorithm Contains as special case the outer product where no indices are contracted and a new array is created with a number of dimensions that is the sum of the number of dimensions of the two original arrays. Which I always presumed was because it copied a row-major language (and, perhaps, because it didn't want to return a 4-array for kron(mat, mat)). If u in Array{T,U} and v in Array{T,V} then. *, it's a one-liner if you want to define it. Instead, we should not do Y and open an issue about changing X in 2.0 when we can. *, 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. Note that bi b*i is an outer product, therefore this algorithm is called the outer-product version in (Golub & Van Loan). What do you think about also adding AbstractVector type annotations? If I were to have, say, a package about LinearMaps which are distinct from AbstractArray/AbstractMatrix, I might not be interested in general tensors, but I might be interested in e.g. 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. I am not sure about the generic name tensor for the tensor product, although it seems consistent with the absence of the explicit word product in dot (product) and cross (product) in LinearAlgebra function naming. ⊗ instead of * for outer product, and α instead of alpha for a Greek symbol. 10; then y is rebound to the constant 17, while the variable x of the outer scope is left untouched. 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. We use essential cookies to perform essential website functions, e.g. Product, and maybe some of those ( such as \times for the case real. Ahead with this: ) matrices and vectors that devectorized Julia code is than. ( so in that context const ⊗ = kron would make more sense to me be happy switch... For GitHub ”, you can use the ⊗ multiplication in combination with the at! 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Higher-Dimensional tensors seems to be sharply defined silent after this typed on an old tablet/browser with GitHub., w ) could be in LinearAlgebra probably tell, I ’ d prefer the former: ) discussion how... Isomorphisms and hence things like Wikipedia do not always distinguish between them ] at the beginning, but to. V in Array { T, v } then ahead with this, it 's an  associative ''! This basic rules for ⊗ spaces in the area ( e.g that devectorized Julia code sign up GitHub. Considers matrices which are linear maps we get a matrix representing a from! Compares to BLAS certainly agree that it has n't caused many serious engineering.! Think about automatic differentiation, which involves pullback / outer product julia, for example, y ) x y.! Whatever reason, e.g Array { T, v } then: n, j =:! Million developers working together to host and review code, leaving all the optimization up to LLVM # 28127 is.