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The Matrix Calculus You Need for Deep Learning
The Matrix Calculus You Need For Deep Learning Terence Parr and Jeremy Howard July 3, 2018 (We teach in University of San Francisco's MS in Data Science program and have other nefarious projects underway. You might know Terence as the creator of the ANTLR parser generator. For more material, see Jeremy's fast.ai courses and University of San Francisco's Data Institute in- person version of the deep learning course.) HTML version (The PDF and HTML were generated from markup using bookish) Abstract This paper is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. We assume no math knowledge beyond what you learned in calculus 1, and provide links to help you refresh the necessary math where needed. Note that you do not need to understand this material before you start learning to train and use deep learning in practice; rather, this material is for those who are already familiar with the basics of neural networks, and wish to deepen their understanding of the underlying math. Don't worry if you get stuck at some point along the way|just go back and reread the previous section, and try writing down and working through some examples. And if you're still stuck, we're happy to answer your questions in the Theory category at forums.fast.ai. Note: There is a reference section at the end of the paper summarizing all the key matrix calculus rules and terminology discussed here. arXiv:1802.01528v3 [cs.LG] 2 Jul 2018 1 Contents 1 Introduction 3 2 Review: Scalar derivative rules4 3 Introduction to vector calculus and partial derivatives5 4 Matrix calculus 6 4.1 Generalization of the Jacobian . -
Lecture 9: Partial Derivatives
Math S21a: Multivariable calculus Oliver Knill, Summer 2016 Lecture 9: Partial derivatives ∂ If f(x,y) is a function of two variables, then ∂x f(x,y) is defined as the derivative of the function g(x) = f(x,y) with respect to x, where y is considered a constant. It is called the partial derivative of f with respect to x. The partial derivative with respect to y is defined in the same way. ∂ We use the short hand notation fx(x,y) = ∂x f(x,y). For iterated derivatives, the notation is ∂ ∂ similar: for example fxy = ∂x ∂y f. The meaning of fx(x0,y0) is the slope of the graph sliced at (x0,y0) in the x direction. The second derivative fxx is a measure of concavity in that direction. The meaning of fxy is the rate of change of the slope if you change the slicing. The notation for partial derivatives ∂xf,∂yf was introduced by Carl Gustav Jacobi. Before, Josef Lagrange had used the term ”partial differences”. Partial derivatives fx and fy measure the rate of change of the function in the x or y directions. For functions of more variables, the partial derivatives are defined in a similar way. 4 2 2 4 3 2 2 2 1 For f(x,y)= x 6x y + y , we have fx(x,y)=4x 12xy ,fxx = 12x 12y ,fy(x,y)= − − − 12x2y +4y3,f = 12x2 +12y2 and see that f + f = 0. A function which satisfies this − yy − xx yy equation is also called harmonic. The equation fxx + fyy = 0 is an example of a partial differential equation: it is an equation for an unknown function f(x,y) which involves partial derivatives with respect to more than one variables. -
A Brief Tour of Vector Calculus
A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0. -
Introduction to Shape Optimization
Introduction to Shape optimization Noureddine Igbida1 1Institut de recherche XLIM, UMR-CNRS 6172, Facult´edes Sciences et Techniques, Universit´ede Limoges 123, Avenue Albert Thomas 87060 Limoges, France. Email : [email protected] Preliminaries on PDE 1 Contents 1. PDE ........................................ 2 2. Some differentiation and differential operators . 3 2.1 Gradient . 3 2.2 Divergence . 4 2.3 Curl . 5 2 2.4 Remarks . 6 2.5 Laplacian . 7 2.6 Coordinate expressions of the Laplacian . 10 3. Boundary value problem . 12 4. Notion of solution . 16 1. PDE A mathematical model is a description of a system using mathematical language. The process of developing a mathematical model is termed mathematical modelling (also written model- ing). Mathematical models are used in many area, as in the natural sciences (such as physics, biology, earth science, meteorology), engineering disciplines (e.g. computer science, artificial in- telligence), in the social sciences (such as economics, psychology, sociology and political science); Introduction to Shape optimization N. Igbida physicists, engineers, statisticians, operations research analysts and economists. Among this mathematical language we have PDE. These are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic. In this section, we give some basic example of elliptic partial differential equation (PDE) of second order : standard Laplacian and Laplacian with variable coefficients. -
High Order Gradient, Curl and Divergence Conforming Spaces, with an Application to NURBS-Based Isogeometric Analysis
High order gradient, curl and divergence conforming spaces, with an application to compatible NURBS-based IsoGeometric Analysis R.R. Hiemstraa, R.H.M. Huijsmansa, M.I.Gerritsmab aDepartment of Marine Technology, Mekelweg 2, 2628CD Delft bDepartment of Aerospace Technology, Kluyverweg 2, 2629HT Delft Abstract Conservation laws, in for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. We develop high order interpolants, from any basis that is a partition of unity, that satisfy these integral relations exactly, at cell level. The resulting gradient, curl and divergence conforming spaces have the propertythat the conservationlaws become completely independent of the basis functions. This means that the conservation laws are exactly satisfied even on curved meshes. As an example, we develop high ordergradient, curl and divergence conforming spaces from NURBS - non uniform rational B-splines - and thereby generalize the compatible spaces of B-splines developed in [1]. We give several examples of 2D Stokes flow calculations which result, amongst others, in a point wise divergence free velocity field. Keywords: Compatible numerical methods, Mixed methods, NURBS, IsoGeometric Analyis Be careful of the naive view that a physical law is a mathematical relation between previously defined quantities. The situation is, rather, that a certain mathematical structure represents a given physical structure. Burke [2] 1. Introduction In deriving mathematical models for physical theories, we frequently start with analysis on finite dimensional geometric objects, like a control volume and its bounding surfaces. We assign global, ’measurable’, quantities to these different geometric objects and set up balance statements. -
Vector Calculus and Differential Forms with Applications To
Vector Calculus and Differential Forms with Applications to Electromagnetism Sean Roberson May 7, 2015 PREFACE This paper is written as a final project for a course in vector analysis, taught at Texas A&M University - San Antonio in the spring of 2015 as an independent study course. Students in mathematics, physics, engineering, and the sciences usually go through a sequence of three calculus courses before go- ing on to differential equations, real analysis, and linear algebra. In the third course, traditionally reserved for multivariable calculus, stu- dents usually learn how to differentiate functions of several variable and integrate over general domains in space. Very rarely, as was my case, will professors have time to cover the important integral theo- rems using vector functions: Green’s Theorem, Stokes’ Theorem, etc. In some universities, such as UCSD and Cornell, honors students are able to take an accelerated calculus sequence using the text Vector Cal- culus, Linear Algebra, and Differential Forms by John Hamal Hubbard and Barbara Burke Hubbard. Here, students learn multivariable cal- culus using linear algebra and real analysis, and then they generalize familiar integral theorems using the language of differential forms. This paper was written over the course of one semester, where the majority of the book was covered. Some details, such as orientation of manifolds, topology, and the foundation of the integral were skipped to save length. The paper should still be readable by a student with at least three semesters of calculus, one course in linear algebra, and one course in real analysis - all at the undergraduate level. -
1 Space Curves and Tangent Lines 2 Gradients and Tangent Planes
CLASS NOTES for CHAPTER 4, Nonlinear Programming 1 Space Curves and Tangent Lines Recall that space curves are de¯ned by a set of parametric equations, x1(t) 2 x2(t) 3 r(t) = . 6 . 7 6 7 6 xn(t) 7 4 5 In Calc III, we might have written this a little di®erently, ~r(t) =< x(t); y(t); z(t) > but here we want to use n dimensions rather than two or three dimensions. The derivative and antiderivative of r with respect to t is done component- wise, x10 (t) x1(t) dt 2 x20 (t) 3 2 R x2(t) dt 3 r(t) = . ; R(t) = . 6 . 7 6 R . 7 6 7 6 7 6 xn0 (t) 7 6 xn(t) dt 7 4 5 4 5 And the local linear approximation to r(t) is alsoRdone componentwise. The tangent line (in n dimensions) can be written easily- the derivative at t = a is the direction of the curve, so the tangent line is given by: x1(a) x10 (a) 2 x2(a) 3 2 x20 (a) 3 L(t) = . + t . 6 . 7 6 . 7 6 7 6 7 6 xn(a) 7 6 xn0 (a) 7 4 5 4 5 In Class Exercise: Use Maple to plot the curve r(t) = [cos(t); sin(t); t]T ¼ and its tangent line at t = 2 . 2 Gradients and Tangent Planes Let f : Rn R. In this case, we can write: ! y = f(x1; x2; x3; : : : ; xn) 1 Note that any function that we wish to optimize must be of this form- It would not make sense to ¯nd the maximum of a function like a space curve; n dimensional coordinates are not well-ordered like the real line- so the fol- lowing statement would be meaningless: (3; 5) > (1; 2). -
Integrating Gradients
Integrating gradients 1 dimension The \gradient" of a function f(x) in one dimension (i.e., depending on only one variable) is just the derivative, f 0(x). We want to solve f 0(x) = k(x); (1) where k(x) is a known function. When we find a primitive function K(x) to k(x), the general form of f is K plus an arbitrary constant, f(x) = K(x) + C: (2) Example: With f 0(x) = 2=x we find f(x) = 2 ln x + C, where C is an undetermined constant. 2 dimensions We let the function depend on two variables, f(x; y). When the gradient rf is known, we have known functions k1 and k2 for the partial derivatives: @ f(x; y) = k (x; y); @x 1 @ f(x; y) = k (x; y): (3) @y 2 @f To solve this, we integrate one of them. To be specific, we here integrate @x over x. We find a primitive function with respect to x (thus keeping y constant) to k1 and call it k3. The general form of f will be k3 plus an arbitrary term whose x-derivative is zero. In other words, f(x; y) = k3(x; y) + B(y); (4) where B(y) is an unknown function of y. We have made some progress, because we have replaced an unknown function of two variables with another unknown function depending only on one variable. @f The best way to come further is not to integrate @y over y. That would give us a second unknown function, D(x). -
Mean Value Theorem on Manifolds
MEAN VALUE THEOREMS ON MANIFOLDS Lei Ni Abstract We derive several mean value formulae on manifolds, generalizing the clas- sical one for harmonic functions on Euclidean spaces as well as the results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to `heat spheres' is proved for heat equation with respect to evolving Riemannian metrics via a space-time consideration. Some new monotonicity formulae are derived. As applications of the new local monotonicity formulae, some local regularity theorems concerning Ricci flow are proved. 1. Introduction The mean value theorem for harmonic functions plays an central role in the theory of harmonic functions. In this article we discuss its generalization on manifolds and show how such generalizations lead to various monotonicity formulae. The main focuses of this article are the corresponding results for the parabolic equations, on which there have been many works, including [Fu, Wa, FG, GL1, E1], and the application of the new monotonicity formula to the study of Ricci flow. Let us start with the Watson's mean value formula [Wa] for the heat equation. Let U be a open subset of Rn (or a Riemannian manifold). Assume that u(x; t) is 2 a C solution to the heat equation in a parabolic region UT = U £ (0;T ). For any (x; t) de¯ne the `heat ball' by 8 9 jx¡yj2 < ¡ 4(t¡s) = e ¡n E(x; t; r) := (y; s) js · t; n ¸ r : : (4¼(t ¡ s)) 2 ; Then Z 1 jx ¡ yj2 u(x; t) = n u(y; s) 2 dy ds r E(x;t;r) 4(t ¡ s) for each E(x; t; r) ½ UT . -
Using Surface Integrals for Checking the Archimedes' Law of Buoyancy
Using surface integrals for checking the Archimedes’ law of buoyancy F M S Lima Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasilia-DF, Brazil E-mail: [email protected] Abstract. A mathematical derivation of the force exerted by an inhomogeneous (i.e., compressible) fluid on the surface of an arbitrarily-shaped body immersed in it is not found in literature, which may be attributed to our trust on Archimedes’ law of buoyancy. However, this law, also known as Archimedes’ principle (AP), does not yield the force observed when the body is in contact to the container walls, as is more evident in the case of a block immersed in a liquid and in contact to the bottom, in which a downward force that increases with depth is observed. In this work, by taking into account the surface integral of the pressure force exerted by a fluid over the surface of a body, the general validity of AP is checked. For a body fully surrounded by a fluid, homogeneous or not, a gradient version of the divergence theorem applies, yielding a volume integral that simplifies to an upward force which agrees to the force predicted by AP, as long as the fluid density is a continuous function of depth. For the bottom case, this approach yields a downward force that increases with depth, which contrasts to AP but is in agreement to experiments. It also yields a formula for this force which shows that it increases with the area of contact. PACS numbers: 01.30.mp, 01.55.+b, 47.85.Dh Submitted to: Eur. -
Gradient Sparsification for Communication-Efficient Distributed
Gradient Sparsification for Communication-Efficient Distributed Optimization Jianqiao Wangni Jialei Wang University of Pennsylvania Two Sigma Investments Tencent AI Lab [email protected] [email protected] Ji Liu Tong Zhang University of Rochester Tencent AI Lab Tencent AI Lab [email protected] [email protected] Abstract Modern large-scale machine learning applications require stochastic optimization algorithms to be implemented on distributed computational architectures. A key bottleneck is the communication overhead for exchanging information such as stochastic gradients among different workers. In this paper, to reduce the communi- cation cost, we propose a convex optimization formulation to minimize the coding length of stochastic gradients. The key idea is to randomly drop out coordinates of the stochastic gradient vectors and amplify the remaining coordinates appropriately to ensure the sparsified gradient to be unbiased. To solve the optimal sparsification efficiently, a simple and fast algorithm is proposed for an approximate solution, with a theoretical guarantee for sparseness. Experiments on `2-regularized logistic regression, support vector machines and convolutional neural networks validate our sparsification approaches. 1 Introduction Scaling stochastic optimization algorithms [26, 24, 14, 11] to distributed computational architectures [10, 17, 33] or multicore systems [23, 9, 19, 22] is a crucial problem for large-scale machine learning. In the synchronous stochastic gradient method, each worker processes a random minibatch of its training data, and then the local updates are synchronized by making an All-Reduce step, which aggregates stochastic gradients from all workers, and taking a Broadcast step that transmits the updated parameter vector back to all workers. -
Computer Problems for Vector Calculus
Chapter: Vector Calculus Computer Problems for Vector Calculus 2 2 1. The average rainfall in Flobbertown follows the strange pattern R = (1 + sin x) e−x y , where x and y are distances north and east from one corner of the town. (a) Pick at least 2 values of x and sketch the rainfall function R(y) at that value. Label your plots with their x values. (b) Pick at least 2 values of y and sketch the rainfall function R(x) at that value. Label your plots with their x values. (c) Have a computer generate a 3D plot of R(x; y). Make sure you plot over a region that clearly shows the general behavior of the function, and includes all the x and y values you used for your sketches. Check if the computer plot matches your constant-latitude and constant-longitude sketches. (If it doesn't, figure out what you did wrong.) (d) Based on your computer plot, if you were to start at the position (π=4; 1) roughly what direction could you move in to keep R constant? Hint: You may find it easier to answer this if you make a second plot that zooms in on a small region around this point. 2. The town of Chewandswallow has been buried in piles of bread. The depth of bread is given by B = cos(x + y) + sin x2 + y2, where the town covers the region 0 ≤ x ≤ π=2, 0 ≤ y ≤ π. (a) Have a computer make a contour plot of B(x; y).