Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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1answer
48 views

How to understand mapping function of kernel?

For a kernel function, we have two conditions one is that it should be symmetric which is easy to understand intuitively because dot products are symmetric as well and our kernel should also follow ...
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1answer
44 views

What are the variables used in a Gaussian radial basis kernel in the context of SVMs?

If I have the Gaussian kernel $$ k(x, x') = \operatorname{exp}\left( -\| x - x' \|^2 / 2\sigma^2 \right) $$ What is $x$ and $x'$ in the context of training an SVM?
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1answer
183 views

Does TD(0) prediction require Robbins-Monro conditions to converge to the value function?

Does the learning rate parameter $\alpha$ require the Robbins-Monro conditions below for the TD(0) algorithm to converge to the true value function of a policy? $$\sum \alpha_t =\infty \quad \text{...
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1answer
93 views

Why does the discriminator minimize the cross-entropy while the generator maximize it?

In his original GAN paper Goodfellow gives a game theoretic perspective for GANs: \begin{equation} \underset{G}{\min}\, \underset{D}{\max}\, V\left(D,G \right) = \mathbb{E}_{x\sim\mathit{p}_{\...
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3answers
92 views

Is it possible to compute $P( F \mid S )$ given $P(F \mid S,A)$, $P(F \mid S, \lnot A)$?

I have a bayesian network, which has the following data: $P(S) = 0.07$ $P(A) = 0.01$ $P(F \mid S,A) = 1.0$ $P(F \mid S, \lnot A) = 0.7$ $P(F \mid \lnot S, A) = 0.9$ $P(F \mid \lnot S, \lnot A) =...
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1answer
67 views

Is the next state drawn from the joint distribution of the previous state and action?

Suppose $G_t$, the discounted return at time $t$ is defined as: $$ G_t \triangleq R_t+\gamma R_{t+1}+\gamma^{2}R_{t+2} + \cdots = \sum_{j=1}^{\infty} \gamma^{k}R_{t+k}$$ where $R_t$ is the reward at ...
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1answer
175 views

How are vectors and matrices multiplied in supervised machine learning?

I've recently started reading a book about deep learning. The book is titled "Grokking Deep Learning" (by Andrew W Trask). In chapter 3 (pages 44 and 45), it talks about multiplying vectors using dot ...
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0answers
29 views

Mathematically speaking, Is it only the product operation used in the chain rule causing the vanishing or exploding gradient?

I am asking this question from the mathematical perspective of the vanishing and exploding gradient problems that we face generally during training deep neural networks. The chain rule of ...
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0answers
30 views

Is it true that real world data is highly discontinuous?

A function $f$ is said to be continuous at a point $c$ if it satisfies three properties: Should be defined at the point $c$ Left and right-hand limits at $c$ must be equal i.e., the limit must exist ...
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0answers
16 views

Is there any concept like 'applying affine transformation on multiple inputs'?

Affine transformation on $X$ is a transformation of the following form $$Y = wX + b$$ In general, $w, X, Y$ and $b$ tensors. We generally call tensor $X$ as an input to affine transformation or the ...
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0answers
32 views

How can I compute a mathematical formula for my CNN?

Let's say, for example, I have built the following CNN model using Keras: ...
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0answers
33 views

Could the inputs of the mean squared-error loss function be transformed to allow larger learning rates?

In the context of a neural network $\hat{y} = f_\theta(\mathbf{x})$ with parameters $\theta$ that is trained to perform regression such that the prediction $\hat{\mathbf{y}} = [\hat{y}_1,\hat{y}_2,...,...
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0answers
18 views

Clonal operator in Immune Clonal Strategy

I was reading about Immune Clonal Strategy, specifically about Monoclonal operator from Immunity clonal strategies, and it goes as follows: Here $a_i $ is a point and $a_i = \{ x_1, x_2, \cdots, x_m \...
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1answer
54 views

Is there a full and precise formulation of Theorem 1 in the Integrated Gradients paper?

Theorem 1 (page 5) in the paper about Integrated Gradients states that Integrated gradients is the unique path method that is symmetry-preserving. What I miss is A precise formulation of the ...
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0answers
33 views

Are monotonically increasing functions easier to learn?

A monotonically increasing function is a function that as x gets bigger so does its output. So, if plotted, it will never go down. Although the outputs might stay constant. Logically this seems like ...
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1answer
75 views

Explanation of this L2 minimization equation

I am trying to understand the last two lines of this math notation (from this paper). How did Var and double summation of Cov come to the equation? The first two lines I understood something like $(a-...
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0answers
35 views

Can any area of math come into play in Machine Learning Research?

As I read online following areas in mathematics comes into play in ML research Linear Algebra Calculus Differential Equations Probability Statistics Discrete Mathematics Optimization Analytic ...
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0answers
38 views

How do I derive the gradient of the log-likelihood of an RBM?

In a Restricted Boltzmann Machine (RBM), the likelihood function is: $$p(\mathbf{v};\mathbf{\theta}) = \frac{1}{Z} \sum_{\mathbf{h}} e^{-E(\mathbf{v},\mathbf{h};\mathbf{\theta})}$$ Where $E$ is the ...
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0answers
11 views

How do I find the data-point with respect to a given frame?

I've been reading this paper that formulates invariant task-parametrized HSMMs. In section 3.1 (Model Learning), the task parameters are represented in $F$ coordinate systems defined by $\{A_j,b_j\}_{...
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0answers
45 views

Could the neural network automatically calculate and get different one-to-many quantities relative to their parent quantity?

Let's say I have a primary dataset that its secondary dataset is hundreds to match and group like an one-to-many relationship. I'm new in this world of the AI but my problem is that many child groups ...
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0answers
200 views

What is an auto-associator?

What is an auto-associator, and how does it work? How can we design an auto-associator for a given pattern? I couldn't find a clear explanation for this anywhere on the internet. Here's an example of ...
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0answers
44 views

Simplifying Log Loss

I am reading through a paper (https://www.mitpressjournals.org/doi/pdf/10.1162/0891201053630273) where they describe logloss as a ranking function and can be simplified to the margin of the training ...
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0answers
35 views

How do I approach this problem?

Let's say I have a dataset with multiple types of multiple ingredients (salt1,salt2, etc). Each n-th variation of each ingredient vs flavor may be represented by an n×k matrix that where an ingredient ...
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0answers
22 views

How do I decide which norm to use for placing a constraint on my adversarial perturbation?

I am performing an adversarial machine learning attack on a neural network for network traffic classification. For adding adversarial perturbations in features such as packet interarrival times and ...
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1answer
335 views

In a single neuron output layer should the output be a scalar?

Given a neural network with 3 inputs, 4 hidden layers, and 1 output, should the output neuron be a vector or a scalar? I thought that at the end of the summation only one number between 0 and 1 would ...
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1answer
44 views

What is the equation of the separation line for this neuron with identity activation?

I have a single neuron with 2 inputs, and identity activation, where f is activation function and u is output: $u = f(w_1x_1 + ...
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2answers
226 views

Formal proof that every purely reactive agent has behaviorally equivalent standard agent

It kind of makes sense intuitively but I'm not sure about a formal proof. I'll start with briefly listing definitions from Intro to Multiagent systems, Wooldridge, 2002 and then give you my reasoning ...
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1answer
84 views

Given an axis-angle rotation vector, how can I find the unit rotation axis and angle?

I have a robotics assignment, which I am unable to solve. Given the axis-angle rotation vector $\Theta = (2, 2, 0)$, how can I calculate the unit vector of the rotation axis $k$ and the angle $\theta$?...
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0answers
41 views

Where does the expectation term in the derivative of the soft-max policy come from?

At slide 17 of the David Silver's series, the soft-max policy is defined as follows $$ \pi_\theta(s, a) \propto e^{\phi(s, a)^T \theta} $$ that is, the probability of an action $a$ (in state $s$) is ...
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0answers
26 views

Since there are different types of neurons in adjacent positions in the brain's arrays, should heterogeneous layers be developed?

Below is a taxonomy of neurons. Some of these types occur in different locations in the brain, but there are adjacent neurons of varying types with clearly functional type diversity in many parts of ...
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2answers
233 views

In the context of importance sampling ratio, how is the equation $\mathbb{E}\left[\rho_{t: T-1} G_{t} | S_{t}=s\right]=v_{\pi}(s)$ derived?

When reading the book by Sutton and Barto, I came across the importance sampling ratio. The first equation, I believe, describes the probability a particular sequence is obtained given the current ...
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0answers
55 views

Optimization step in Apprenticeship Learning via Inverse Reinforcement Learning

Why the optimization step of the algorithm a quadratic program? [See: Apprenticeship Learning via Inverse Reinforcement Learning; page 3] Isn't the objective function linear? Why don't we treat ...
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0answers
77 views

How does this sigma work?(Harris algorithm) [closed]

May someone explains some first iterations of this sigma? Also, how did it convert the above expression to below expression? What it the meaning of I(x) and I(y)?
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2answers
33 views

Reason for relaxing limit in derivative in this context?

Consider the following paragraph from NUMERICAL COMPUTATION of the deep learning book.. Suppose we have a function $y = f(x)$, where both $x$ and $y$ are real numbers. The derivative of this function ...
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1answer
32 views

How to calculate the gradient penalty proposed in "Improved Training of Wasserstein GANs"?

The research paper titled Improved Training of Wasserstein GANs proposed a gradient penalty in order to avoid undesired behavior due to weight clipping of the discriminator. We now propose an ...
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2answers
66 views

What is meant by an axis of a tensor?

Tensor is an ordered collection of elements. The elements are generally real numbers. Tensors are used in deep learning for storing data. There is a wide usage of the word "axis" related to ...
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1answer
123 views

What is Bayes' theorem?

What is Bayes' theorem? How does it relate to conditional probabilities?
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1answer
88 views

In this paper, if region $R_{2}$ moves in a sliding window manner, won't the saliency map have a smaller size than the original image?

In the paper Salient Region Detection and Segmentation, I have a question pertaining to section 3 on the convolution-like operation being performed. I had already asked a few questions about the paper ...
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1answer
26 views

Isssue in understanding the derivation regarding mean squared error

The following derivation is taken from Chapter 5: Machine Learning Basics from the book titled Deep Learning (by Aaron Courville et al.) I am facing difficulty in understanding the zero derivative ...
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1answer
33 views

What are the Calculus books recommended for begineer to advanced researchers in artificial intelligence?

Calculus is a branch of mathematics that deals with the rate of change of outputs of a function w.r.t the inputs. It contains several concepts including limits, first-order derivatives, higher-order ...
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1answer
32 views

How to interpret the policy gradient expression in reinforcement learning?

I'm currently going through the OpenAI's spinning up introduction course to reinforcement learning. On one of the final sections, they derive an expression for the gradient of the undiscounted return ...
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1answer
50 views

What's the difference between a 1d tensor and a 2d tensor with 1 dimension?

I'm doing a TensorFlow tutorial, where they convert an array of the numbers [1,2,3] to a tensor like this: ...
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1answer
58 views

Do the rows of the design matrix refer to the observations or predictors?

I attempt to understand the formulation of dictionary learning for this paper: Depression Detection via Harvesting Social Media: A Multimodal Dictionary Learning Solution Multimodal Task-Driven ...
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1answer
90 views

Mathematical calculation behind decision tree classifier with continuous variables

I am working on a binary classification problem having continuous variables (Gene expression Values). My goal is to classify the samples as case or ...
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1answer
47 views

Why are all weights of a neural net updated and not just the weights of the first layer

Why are all weights of a neural net updated and not only the weights of the first hidden layer? The error-influence of the prediction by the weights of a neural net is calculated using the chain rule....
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1answer
67 views

How to understand the average l2 loss?

In the snippet below, the highlighted part is the average norm, but since $1/|p_i|$ is outside the summation, it is very confusing to understand. is $|p_i|$ l2-norm(as per wolfram) or l1-norm or ...
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0answers
82 views

Notations and definitions relative to distributional reinforcement learning [closed]

I am reading the paper A Distributional Perspective on Reinforcement Learning, Marc G. Bellemare et al. The vague notation in this paper is very annoying. I searched several papers to solve this ...
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0answers
67 views

What's wrong with my understanding of how RNNs work?

Recently, I've been trying to derive the mathematics behind various Neural Network structures. I managed to derive the MLP and tested it to be on par with a Keras implementation (Using the MNIST ...
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0answers
23 views

What are the different types of geometry in literature that may be used for deep learning?

Recently, I asked a question on the concept of a manifold and received an answer that points to a relatively new subfield of deep learning named geometric deep learning. In the ...
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0answers
36 views

What are mathematically the factors of variation in deep learning?

The following paragraph from an answer tells us about factors of variation Factors of variation are some factors which determine varieties in observed data. If that factors change, the behaviour of ...