I"m spring at some formulas including matrices (in the context of machine learning, but I"m not certain it"s relevant) and also I came throughout $odot$. What might this mean? The context is $M odot N$, whereby $M$ is a matrix and also $N$ can be a vector, or a matrix, or a scalar, it"s a bit thick so it"s difficult to tell. I have reason to believe it may be the Hadamard product, is there anything rather it might mean?


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In the LSTM equations, the circled dot operator is typically used to stand for element-wise multiplication.

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When i researched around the prize ⊙. I acquired two things:

$1$. In the publication Meaning, Logic and Ludics-By Alain Lecomte, The writer says that: let $U$ and $B$ be two confident designs we signify tensor product that $U$ and $B$ by $U$⊙$B$.

$2$. In the book Dag Prawitz top top Proofs and Meaning: The writer says that $alpha⊙eta$ means that one of two people $eta $ is derivable indigenous $alpha$ or is $alpha$ derivable from $eta $.

I think that second on e renders sense, as for tensor product ns have constantly seen the prize ⊗ gift used. I provided you the info which ns had, expect it helps.


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answered Dec 2 "16 in ~ 17:01
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Vidyanshu MishraVidyanshu Mishra
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$egingroup$ This is the correct answer! $endgroup$
–user575518
Nov 7 "19 in ~ 10:43
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We use the circle with a dot notation in nonlinear continuum mechanics. The is identified in table of contents notation aseginalignleft(oldsymbolM ⊙ oldsymbolN ight)_ABCD= dfrac12left(M_ACN_BD+M_ADN_BC ight)endalignwhere $oldsymbolM$ and $oldsymbolN$ are second order tensors. The result of this procedure is a 4th order tensor. You can think of the as concerned the outer product of 2 matrices or $oldsymbolMotimes oldsymbolN$ in the it take away two 2nd order tensors and generates a fourth order tensor. Remember the external product is identified for two 2nd order tensors as

eginalignleft(oldsymbolM otimes oldsymbolN ight)_ABCD= M_ABN_CDendalign

The circle with a dot procedure $⊙$ occurs once calculating the elastic modulus $lungemine.combfunderlineC$ (a fourth order tensor) from constitutive laws. Examples encompass the Mooney-Rivlin or Neohookean product models. To acquire the elastic modulus you need to take derivatives of tensors v respect to various other tensors. Because that instance,

eginalignunderlinelungemine.combfC = 2dfracdoldsymbolSdoldsymbolCendalignor in index notation aseginalign extC_ABCD = 2dfracpartial S_ABpartial C_CDendalignwhere $oldsymbolC$ is the left Cauchy-Green deformation tensor and $oldsymbolS$ is the 2nd Piola-Kirchhoff anxiety tensor. Based on the constitutive models, i m sorry relate stress in a product to the strain, it turns out $oldsymbolS$ is a duty of the train station of $oldsymbolC$. If you occupational it the end you will discover that girlfriend will need to compute $fracpartial oldsymbolC^-1partial oldsymbolC$. The proof is a headache yet the result comes the end toeginaligndfracpartial oldsymbolC^-1partial oldsymbolC = -oldsymbolC^-1 ⊙ oldsymbolC^-1endalignIn index notationeginaligndfracpartial C^-1_ABpartial C_CD = -dfrac12left(C^-1_ACC^-1_BD+C^-1_ADC^-1_BC ight)=-left(oldsymbolC^-1 ⊙ oldsymbolC^-1 ight)_ABCDendalignThe circle with a dot operation only arises due to the fact that $oldsymbolC$ is a symmetric matrix, i.e., $oldsymbolC=oldsymbolC^T$ and $oldsymbolC_sym=dfrac12left(oldsymbolC+oldsymbolC^T ight) = oldsymbolC$. Keep in mind that if acquisition the derivative that an inverse of a nonsymmetric tensor with respect to itself yieldseginaligndfracpartial A_AB^-1partial A_CD=-A^-1_ACA^-1_DBendalign and also this is no the external product. This operation has not however been provided a symbol.

Note:

the outer product $otimes$ is also called the tensor product.The exponentiation are capital letters ABCD since in continuum mechanics resources letters signify Lagrangian/material (or recommendation configuration) coordinates. Lower situation indices ijkl denote spatial/Eulerian coordinates.

Addendum:Other uses I have actually seen for $⊙$ include

In physics, I have actually seen it typical a point resource such together a suggest charge or gravity resource like a planet.In physics, I have actually seen it average the vector point out out that the page $⊙$. And also $otimes$ method the direction the the vector is into the page. I have actually seen this in E&M because that B-fields and E-fields and mechanics for torques.In lungemine.com it might mean a role composition operator, i m sorry maps functions to functions, e.g., $,f⊙g$.

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This is what ns think it is provided for in that long Short-Term storage neural networking great https://www.youtube.com/watch?v=iX5V1WpxxkY&feature=youtu.be at 46:00.

In lungemine.comematics, practical compositions room usually denoted by a small circle $circ$. Because that example, Eulerian $f(oldsymbolx,t)$ and Lagrangian $F(oldsymbolX,t)$ explanation are regarded each various other by a duty composition:eginalignF(oldsymbolX,t)=f(oldsymbolPhi(oldsymbolX,t),t) extrm or F = f circ oldsymbolPhiendalign