Machine Learning

Andrew Rosenberg

Lecture 3: Math Primer II

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Today

• Wrap up of probability• Vectors, Matrices.• Calculus• Derivation with respect to a vector.

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Properties of probability density functions

Sum Rule

Product Rule

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Expected Values

• Given a random variable, with a distribution p(X), what is the expected value of X?

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Multinomial Distribution

• If a variable, x, can take 1-of-K states, we represent the distribution of this variable as a multinomial distribution.

• The probability of x being in state k is μk

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Expected Value of a Multinomial

• The expected value is the mean values.

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Gaussian Distribution

• One Dimension

• D-Dimensions

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Gaussians

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How machine learning uses statistical modeling

• Expectation– The expected value of a function is the

hypothesis

• Variance– The variance is the confidence in that

hypothesis

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Variance

• The variance of a random variable describes how much variability around the expected value there is.

• Calculated as the expected squared error.

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Covariance

• The covariance of two random variables expresses how they vary together.

• If two variables are independent, their covariance equals zero.

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Linear Algebra

• Vectors– A one dimensional array. – If not specified, assume x is a column

vector.

• Matrices– Higher dimensional array.– Typically denoted with capital letters.– n rows by m columns

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Transposition

• Transposing a matrix swaps columns and rows.

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Transposition

• Transposing a matrix swaps columns and rows.

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Addition

• Matrices can be added to themselves iff they have the same dimensions.– A and B are both n-by-m matrices.

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Multiplication

• To multiply two matrices, the inner dimensions must be the same.– An n-by-m matrix can be multiplied by an m-by-k matrix

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Inversion

• The inverse of an n-by-n or square matrix A is denoted A-1, and has the following property.

• Where I is the identity matrix is an n-by-n matrix with ones along the diagonal.– Iij = 1 iff i = j, 0 otherwise

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Identity Matrix

• Matrices are invariant under multiplication by the identity matrix.

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Helpful matrix inversion properties

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Norm

• The norm of a vector, x, represents the euclidean length of a vector.

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Positive Definite-ness

• Quadratic form– Scalar

– Vector

• Positive Definite matrix M

• Positive Semi-definite

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Calculus

• Derivatives and Integrals• Optimization

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Derivatives

• A derivative of a function defines the slope at a point x.

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Derivative Example

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Integrals

• Integration is the inverse operation of derivation (plus a constant)

• Graphically, an integral can be considered the area under the curve defined by f(x)

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Integration Example

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Vector Calculus

• Derivation with respect to a matrix or vector

• Gradient• Change of Variables with a Vector

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Derivative w.r.t. a vector

• Given a vector x, and a function f(x), how can we find f’(x)?

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Derivative w.r.t. a vector

• Given a vector x, and a function f(x), how can we find f’(x)?

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Example Derivation

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Example Derivation

Also referred to as the gradient of a function.

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Useful Vector Calculus identities

• Scalar Multiplication

• Product Rule

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Useful Vector Calculus identities

• Derivative of an inverse

• Change of Variable

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Optimization

• Have an objective function that we’d like to maximize or minimize, f(x)

• Set the first derivative to zero.

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Optimization with constraints

• What if I want to constrain the parameters of the model.– The mean is less than 10

• Find the best likelihood, subject to a constraint.

• Two functions:– An objective function to maximize– An inequality that must be satisfied

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Lagrange Multipliers

• Find maxima of f(x,y) subject to a constraint.

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General form

• Maximizing:

• Subject to:

• Introduce a new variable, and find a

maxima.

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Example

• Maximizing:

• Subject to:

• Introduce a new variable, and find a

maxima.

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Example

Now have 3 equations with 3 unknowns.

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ExampleEliminate Lambda Substitute and Solve

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Why does Machine Learning need these tools?

• Calculus– We need to identify the maximum likelihood, or

minimum risk. Optimization– Integration allows the marginalization of

continuous probability density functions

• Linear Algebra– Many features leads to high dimensional spaces– Vectors and matrices allow us to compactly

describe and manipulate high dimension al feature spaces.

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Why does Machine Learning need these tools?

• Vector Calculus– All of the optimization needs to be performed

in high dimensional spaces– Optimization of multiple variables

simultaneously – Gradient Descent– Want to take a marginal over high

dimensional distributions like Gaussians.

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Next Time

• Linear Regression– Then Regularization