The Matrix Calculus You Need for Deep Learning
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IVC Factsheet Functions Comp Inverse
Imperial Valley College Math Lab Functions: Composition and Inverse Functions FUNCTION COMPOSITION In order to perform a composition of functions, it is essential to be familiar with function notation. If you see something of the form “푓(푥) = [expression in terms of x]”, this means that whatever you see in the parentheses following f should be substituted for x in the expression. This can include numbers, variables, other expressions, and even other functions. EXAMPLE: 푓(푥) = 4푥2 − 13푥 푓(2) = 4 ∙ 22 − 13(2) 푓(−9) = 4(−9)2 − 13(−9) 푓(푎) = 4푎2 − 13푎 푓(푐3) = 4(푐3)2 − 13푐3 푓(ℎ + 5) = 4(ℎ + 5)2 − 13(ℎ + 5) Etc. A composition of functions occurs when one function is “plugged into” another function. The notation (푓 ○푔)(푥) is pronounced “푓 of 푔 of 푥”, and it literally means 푓(푔(푥)). In other words, you “plug” the 푔(푥) function into the 푓(푥) function. Similarly, (푔 ○푓)(푥) is pronounced “푔 of 푓 of 푥”, and it literally means 푔(푓(푥)). In other words, you “plug” the 푓(푥) function into the 푔(푥) function. WARNING: Be careful not to confuse (푓 ○푔)(푥) with (푓 ∙ 푔)(푥), which means 푓(푥) ∙ 푔(푥) . EXAMPLES: 푓(푥) = 4푥2 − 13푥 푔(푥) = 2푥 + 1 a. (푓 ○푔)(푥) = 푓(푔(푥)) = 4[푔(푥)]2 − 13 ∙ 푔(푥) = 4(2푥 + 1)2 − 13(2푥 + 1) = [푠푚푝푙푓푦] … = 16푥2 − 10푥 − 9 b. (푔 ○푓)(푥) = 푔(푓(푥)) = 2 ∙ 푓(푥) + 1 = 2(4푥2 − 13푥) + 1 = 8푥2 − 26푥 + 1 A function can even be “composed” with itself: c. -
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. -
The Logic of Recursive Equations Author(S): A
The Logic of Recursive Equations Author(s): A. J. C. Hurkens, Monica McArthur, Yiannis N. Moschovakis, Lawrence S. Moss, Glen T. Whitney Source: The Journal of Symbolic Logic, Vol. 63, No. 2 (Jun., 1998), pp. 451-478 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2586843 . Accessed: 19/09/2011 22:53 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. http://www.jstor.org THE JOURNAL OF SYMBOLIC LOGIC Volume 63, Number 2, June 1998 THE LOGIC OF RECURSIVE EQUATIONS A. J. C. HURKENS, MONICA McARTHUR, YIANNIS N. MOSCHOVAKIS, LAWRENCE S. MOSS, AND GLEN T. WHITNEY Abstract. We study logical systems for reasoning about equations involving recursive definitions. In particular, we are interested in "propositional" fragments of the functional language of recursion FLR [18, 17], i.e., without the value passing or abstraction allowed in FLR. The 'pure," propositional fragment FLRo turns out to coincide with the iteration theories of [1]. Our main focus here concerns the sharp contrast between the simple class of valid identities and the very complex consequence relation over several natural classes of models. -
Policy Gradient
Lecture 7: Policy Gradient Lecture 7: Policy Gradient David Silver Lecture 7: Policy Gradient Outline 1 Introduction 2 Finite Difference Policy Gradient 3 Monte-Carlo Policy Gradient 4 Actor-Critic Policy Gradient Lecture 7: Policy Gradient Introduction Policy-Based Reinforcement Learning In the last lecture we approximated the value or action-value function using parameters θ, V (s) V π(s) θ ≈ Q (s; a) Qπ(s; a) θ ≈ A policy was generated directly from the value function e.g. using -greedy In this lecture we will directly parametrise the policy πθ(s; a) = P [a s; θ] j We will focus again on model-free reinforcement learning Lecture 7: Policy Gradient Introduction Value-Based and Policy-Based RL Value Based Learnt Value Function Implicit policy Value Function Policy (e.g. -greedy) Policy Based Value-Based Actor Policy-Based No Value Function Critic Learnt Policy Actor-Critic Learnt Value Function Learnt Policy Lecture 7: Policy Gradient Introduction Advantages of Policy-Based RL Advantages: Better convergence properties Effective in high-dimensional or continuous action spaces Can learn stochastic policies Disadvantages: Typically converge to a local rather than global optimum Evaluating a policy is typically inefficient and high variance Lecture 7: Policy Gradient Introduction Rock-Paper-Scissors Example Example: Rock-Paper-Scissors Two-player game of rock-paper-scissors Scissors beats paper Rock beats scissors Paper beats rock Consider policies for iterated rock-paper-scissors A deterministic policy is easily exploited A uniform random policy -
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. -
Monoids: Theme and Variations (Functional Pearl)
University of Pennsylvania ScholarlyCommons Departmental Papers (CIS) Department of Computer & Information Science 7-2012 Monoids: Theme and Variations (Functional Pearl) Brent A. Yorgey University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/cis_papers Part of the Programming Languages and Compilers Commons, Software Engineering Commons, and the Theory and Algorithms Commons Recommended Citation Brent A. Yorgey, "Monoids: Theme and Variations (Functional Pearl)", . July 2012. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_papers/762 For more information, please contact [email protected]. Monoids: Theme and Variations (Functional Pearl) Abstract The monoid is a humble algebraic structure, at first glance ve en downright boring. However, there’s much more to monoids than meets the eye. Using examples taken from the diagrams vector graphics framework as a case study, I demonstrate the power and beauty of monoids for library design. The paper begins with an extremely simple model of diagrams and proceeds through a series of incremental variations, all related somehow to the central theme of monoids. Along the way, I illustrate the power of compositional semantics; why you should also pay attention to the monoid’s even humbler cousin, the semigroup; monoid homomorphisms; and monoid actions. Keywords monoid, homomorphism, monoid action, EDSL Disciplines Programming Languages and Compilers | Software Engineering | Theory and Algorithms This conference paper is available at ScholarlyCommons: https://repository.upenn.edu/cis_papers/762 Monoids: Theme and Variations (Functional Pearl) Brent A. Yorgey University of Pennsylvania [email protected] Abstract The monoid is a humble algebraic structure, at first glance even downright boring. -
Contents 3 Homomorphisms, Ideals, and Quotients
Ring Theory (part 3): Homomorphisms, Ideals, and Quotients (by Evan Dummit, 2018, v. 1.01) Contents 3 Homomorphisms, Ideals, and Quotients 1 3.1 Ring Isomorphisms and Homomorphisms . 1 3.1.1 Ring Isomorphisms . 1 3.1.2 Ring Homomorphisms . 4 3.2 Ideals and Quotient Rings . 7 3.2.1 Ideals . 8 3.2.2 Quotient Rings . 9 3.2.3 Homomorphisms and Quotient Rings . 11 3.3 Properties of Ideals . 13 3.3.1 The Isomorphism Theorems . 13 3.3.2 Generation of Ideals . 14 3.3.3 Maximal and Prime Ideals . 17 3.3.4 The Chinese Remainder Theorem . 20 3.4 Rings of Fractions . 21 3 Homomorphisms, Ideals, and Quotients In this chapter, we will examine some more intricate properties of general rings. We begin with a discussion of isomorphisms, which provide a way of identifying two rings whose structures are identical, and then examine the broader class of ring homomorphisms, which are the structure-preserving functions from one ring to another. Next, we study ideals and quotient rings, which provide the most general version of modular arithmetic in a ring, and which are fundamentally connected with ring homomorphisms. We close with a detailed study of the structure of ideals and quotients in commutative rings with 1. 3.1 Ring Isomorphisms and Homomorphisms • We begin our study with a discussion of structure-preserving maps between rings. 3.1.1 Ring Isomorphisms • We have encountered several examples of rings with very similar structures. • For example, consider the two rings R = Z=6Z and S = (Z=2Z) × (Z=3Z). -
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 . -
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. -
Math 311 - Introduction to Proof and Abstract Mathematics Group Assignment # 15 Name: Due: at the End of Class on Tuesday, March 26Th
Math 311 - Introduction to Proof and Abstract Mathematics Group Assignment # 15 Name: Due: At the end of class on Tuesday, March 26th More on Functions: Definition 5.1.7: Let A, B, C, and D be sets. Let f : A B and g : C D be functions. Then f = g if: → → A = C and B = D • For all x A, f(x)= g(x). • ∈ Intuitively speaking, this definition tells us that a function is determined by its underlying correspondence, not its specific formula or rule. Another way to think of this is that a function is determined by the set of points that occur on its graph. 1. Give a specific example of two functions that are defined by different rules (or formulas) but that are equal as functions. 2. Consider the functions f(x)= x and g(x)= √x2. Find: (a) A domain for which these functions are equal. (b) A domain for which these functions are not equal. Definition 5.1.9 Let X be a set. The identity function on X is the function IX : X X defined by, for all x X, → ∈ IX (x)= x. 3. Let f(x)= x cos(2πx). Prove that f(x) is the identity function when X = Z but not when X = R. Definition 5.1.10 Let n Z with n 0, and let a0,a1, ,an R such that an = 0. The function p : R R is a ∈ ≥ ··· ∈ 6 n n−1 → polynomial of degree n with real coefficients a0,a1, ,an if for all x R, p(x)= anx +an−1x + +a1x+a0.