latent variable
latent variables are variables that you cannot be observed
70 Matching Annotations
- Nov 2022
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mml-book.github.io mml-book.github.io
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One way to detectoverfitting inpractice is toobserve that themodel has lowtraining risk buthigh test risk duringcross validation
overfitting = high acc during training and low acc during testing
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Model Fitting
how well a model is learning
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cross-validation
technique used to evaluate how well your model is doing
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validation se
I have always been confused by the validation set. It is a set used to provide a glimpse of how your model will react to the data. Usually you take a portion of the training set to create the validation set
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egularization
technique used to reduce overfitting
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overfitting
overfitting = during training the error is small where as during testing it is large
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Another phrasecommonly used forexpected risk is“population risk”
From what I know, population risk is the number of individuals at risk. Is it samething as expected risk ?
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independent and identicallyindependent andidenticallydistributed distributed
what is a set of example here? I am thinking of it as features rather than anything else. But features are dependent upon one another so I am not sure what this means
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Affine functions areoften referred to aslinear functions inmachine learning
affine faction = linear function
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Training or parameter estimation
adjust predictive model based on training data.
In order to find good predictors do one of two things: 1) find the best predict based on some measure of quality (known as finding a point estimate) and 2) using bayesian inference
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Prediction or inference
predict on unseen test data. 'inference' can mean prediction for non-probabilist models or parameter estimation
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goal of learning is to find a model and its corresponding parame-ters such that the resulting predictor will perform well on unseen data
important
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noisy observation
real-life data is always noisy
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example or data point
I thought rows were observations or instances?
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we do not expect the iden-tifier (the Name) to be informative for a machine learning task
This is a good reminder that only query the columns or data that are relevant to the exercise
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features, attributes, or covariates
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What dowe mean by good models?
This is a great question. I usually think of models as algorithms
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good models should perform well on unseendata
The main idea in implementing a machine learning model
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- Oct 2022
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mml-book.github.io mml-book.github.io
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This derivation iseasiest tounderstand bydrawing thereasoning as itprogresses.
The reasoning of the derivative?
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Exampleof a convex set
an easy example to identify convex sets. One way to determine a convex sex is to keep in mind that if at given given point within the set if a line segment is within the set it is convex
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Lagrange multiplier
Lagrange aims to find the local minima and maxima of function
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The step-size is alsocalled the learningrate.
when implementing a neural net, the learning rate is a hyper parameter that controls how much to adjust the weights by w.r.t to the gradient
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We use theconvention of rowvectors forgradients
so a matrix? or just rows like this : [a b c ]?
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minx f (x)
This is important. For all optimization problems, the end goal is to minimize the function
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Linear Program
interesting example. Seeing how the linear programs can be plotted.
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elative frequencies of eventsof interest to the total number of events that occurred
isn't this the definition of mean?
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abducted by aliens
lol
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Theorem 4.3. A square matrix A ∈ Rn×n has det(A) 6 = 0 if and only ifrk(A) = n. In other words, A is invertible if and only if it is full rank
refer to section 2.6.2 for rank definition
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pA(λ) := det(A − λI)
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(−1)k+j det(Ak,j )a cofactor
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det(Ak,j ) is calleda minor
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det(A) =n∑k=1(−1)k+j akj det(Ak,j )
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Adding a multiple of a column/row to another one does not changedet(A)
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Swapping two rows/columns changes the sign of det(A)
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det(λA) = λn det(A)
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If A is regular (invertible), then det(A−1) = 1det(A)
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(4.7)
determinant of 3x3 matrix
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(4.6)
determinant of 2x2 matrix
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is invertibleif and only if det(A) 6 = 0
Invertible: det(A) \(\neq 0\)
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determinant of a square matrix A ∈ Rn×n is a function that maps A
determinant
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- Sep 2022
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mml-book.github.io mml-book.github.io
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rotation matrix
coordinates of rotation in the form on basis vectors
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rotation
linear mapping that rotates a plan by angle \(\theta\) with respect to origin
if angle \(\theta\) > 0 rotate counterclockwise
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orthogonal basis
$$<b_{i}, b_{j}> = 0, i \neq j$$
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orthogonal complement
Let W be a subspace of a vector space V. Then the orthogonal complement of W is also a subspace of V. Furthermore, the intersection of W and its orthogonal complement is just the zero vector.
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normal vector
vector with magnitude 1, \(||w|| = 1\) and is perpendicular to the surface
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Gram-Schmidt process
concatenate basis vector (non-orthogonal and unnormalized) into a matrix, apply gaussian eliminate and obtain an orthonormal basis
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Orthonormal Basi
basis vectors = subset of vectors linearly independent if orthonormal basis -> orthogonal basis
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3.32
distance of orthogonal matrix
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‖Ax‖2 = (Ax)>(Ax) = x>A>Ax = x>Ix = x>x = ‖x‖2
this is an important proof of dot product for an orthogonal matrix
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Orthogonal Matrix
$$AA^T = I = A^TA \Rightarrow A^{-1} = A^T$$ orthonormal columns
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〈x, y〉
this is equal to 1, which does not meet the requirement of orthogonality
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(Orthogonality
if \(<x,y> = 0\) and \(||x|| = ||y|| = 0\)<br /> any two lines that are perpendicular - 90 degree angle
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cos ω
used to find angle between vector
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x, y) 7 → d(x, y)
if x and y are two points in a vector space then, you can find the distance
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d(x, y) := ‖x − y‖ =√〈x − y, x − y
so a Euclidean distance is a distance from point x to point y, so the shortest path (a straight line). I don't understand the difference between distance and Euclidean distance. Isn't distance also a dot product? how would you do the calculation?
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inner product returns smallervalues than the dot product if x1 and x2 have the same sig
this is interesting
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atisfies (3.11) is called symmetric, positive definite
symmetric positive definite
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(3.9)
The inner product must be positive definite, symmetric and bilinear. test for inner product: let v = (1,2) -> <v,v> = (1)(1) - (1)(2) - (2)(1) + 2(2)(2) = 1 - 2 -2 + 8 = -3 + 8 = 5 (symmetric, bilinear and positive)
test for dot product: as per (3.5) the right side does not equal the left side
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se the dot product defined in (3.5), we call(V, 〈·, ·〉) a Euclidean vector space
euclidean vector space
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The pair (V, 〈·, ·〉) is called an inner product space
inner product space
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positive definite, symmetric bilinear mapping Ω : V ×V → R is calledan inner product on V
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positive definite if positive definite∀x ∈ V \{0} : Ω(x, x) > 0 , Ω(0, 0) = 0
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symmetric if Ω(x, y) = Ω(y, x)
a symmetric matrix was: (A^(-1))^T = (A^(T))^-1
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x>y =n∑i=1xiyi
inner product and dot product interchangeable here
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3.4
distance from the origin of a vector
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Positive definite: ‖x‖ > 0 and ‖x‖ = 0 ⇐⇒ x = 0
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Triangle inequality: ‖x + y‖ 6 ‖x‖ + ‖y‖
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Absolutely homogeneous: ‖λx‖ = |λ|‖x‖
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A norm on a vector space V is a function
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