vault backup: 2023-06-07 09:02:27

Affected files:
STEM/AI/Classification/Classification.md
STEM/AI/Classification/Decision Trees.md
STEM/AI/Classification/Gradient Boosting Machine.md
STEM/AI/Classification/Logistic Regression.md
STEM/AI/Classification/Random Forest.md
STEM/AI/Classification/Supervised.md
STEM/AI/Classification/Supervised/README.md
STEM/AI/Classification/Supervised/SVM.md
STEM/AI/Classification/Supervised/Supervised.md
STEM/AI/Learning.md
STEM/AI/Neural Networks/Learning/Boltzmann.md
STEM/AI/Neural Networks/Learning/Competitive Learning.md
STEM/AI/Neural Networks/Learning/Credit-Assignment Problem.md
STEM/AI/Neural Networks/Learning/Hebbian.md
STEM/AI/Neural Networks/Learning/Learning.md
STEM/AI/Neural Networks/Learning/README.md
STEM/AI/Neural Networks/RNN/Autoencoder.md
STEM/AI/Neural Networks/RNN/Deep Image Prior.md
STEM/AI/Neural Networks/RNN/MoCo.md
STEM/AI/Neural Networks/RNN/Representation Learning.md
STEM/AI/Neural Networks/RNN/SimCLR.md
STEM/img/comp-learning.png
STEM/img/competitive-geometric.png
STEM/img/confusion-matrix.png
STEM/img/decision-tree.png
STEM/img/deep-image-prior-arch.png
STEM/img/deep-image-prior-results.png
STEM/img/hebb-learning.png
STEM/img/moco.png
STEM/img/receiver-operator-curve.png
STEM/img/reinforcement-learning.png
STEM/img/rnn+autoencoder-variational.png
STEM/img/rnn+autoencoder.png
STEM/img/simclr.png
STEM/img/sup-representation-learning.png
STEM/img/svm-c.png
STEM/img/svm-non-linear-project.png
STEM/img/svm-non-linear-separated.png
STEM/img/svm-non-linear.png
STEM/img/svm-optimal-plane.png
STEM/img/svm.png
STEM/img/unsup-representation-learning.png

AI/Classification/Classification.md

@@ -9,7 +9,7 @@
 Argument that gives the maximum value from a target function
 
 # Gaussian Classifier
-[Training](Supervised.md)
+[Training](Supervised/Supervised.md)
 - Each class $i$ has it's own Gaussian $N_i=N(m_i,v_i)$
 
 $$\hat i=\text{argmax}_i\left(p(o_t|N_i)\cdot P(N_i)\right)$$

AI/Classification/Decision Trees.md

@@ -0,0 +1,4 @@
+- Flowchart like design
+- Iterative decision making
+
+![](../../img/decision-tree.png)

AI/Classification/Gradient Boosting Machine.md

@@ -0,0 +1,7 @@
+- Higher level take
+- Iteratively train more models addressing weak points
+- Well paired with decision trees
+	- Strictly outperform random forest most of the time
+	- Similar properties
+- One of the best algorithm for dealing with non perceptual data
+- XGBoost

AI/Classification/Logistic Regression.md

@@ -0,0 +1,16 @@
+“hello world”
+Related to naïve bayes
+
+- Statistical model
+- Uses ***logistic function*** to model a ***categorical*** dependent variable
+
+# Types
+- Binary
+	- 2 classes
+- Multinomial
+	- Multiple classes without ordering
+		- Categories
+- Ordinal
+	- Multiple ordered classes
+		- Star rating
+

AI/Classification/Random Forest.md

@@ -0,0 +1 @@
+“Almost always the second best algorithm for any shallow ML task”

AI/Classification/Supervised.md

@@ -1,5 +0,0 @@
-
-# Gaussian Classifier
-- With $T$ labelled data
-
-$$q_t(i)=$$

AI/Classification/Supervised/README.md

@@ -0,0 +1 @@
+Supervised.md

AI/Classification/Supervised/SVM.md

@@ -0,0 +1,74 @@
+[Towards Data Science: SVM](https://towardsdatascience.com/support-vector-machines-svm-c9ef22815589)
+[Towards Data Science: SVM an overview](https://towardsdatascience.com/https-medium-com-pupalerushikesh-svm-f4b42800e989)
+
+- Dividing line between two classes
+	- Optimal hyperplane for a space
+	- Margin maximising hyperplane
+- Can be used for
+	- Classification
+		- SVC
+	- Regression
+		- SVR
+- Alternative to Eigenmodels for supervised classification
+- For smaller datasets
+	- Hard to scale on larger sets
+
+![](../../../img/svm.png)
+- Support vector points
+	- Closest points to the hyperplane
+	- Lines to hyperplane are support vectors
+
+- Maximise margin between classes
+- Take dot product of test point with vector perpendicular to support vector
+- Sign determines class
+
+# Pros
+- Linear or non-linear discrimination
+- Effective in higher dimensions
+- Effective when number of features higher than training examples
+- Best for when classes are separable
+- Outliers have less impact
+
+# Cons
+- Long time for larger datasets
+- Doesn’t do well when overlapping
+- Selecting appropriate kernel
+
+# Parameters
+- C
+	- How smooth the decision boundary is
+	- Larger C makes more curvy
+	- ![](../../../img/svm-c.png)
+- Gamma
+	- Controls area of influence for data points
+	- High gamma reduces influence of faraway points
+
+# Hyperplane
+
+$$\beta_0+\beta_1X_1+\beta_2X_2+\cdot\cdot\cdot+\beta_pX_p=0$$
+- $p$-dimensional space
+- If $X$ satisfies equation
+	- On plane
+- Maximal margin hyperplane
+- Perpendicular distance from each observation to given plane
+	- Best plane has highest distance
+- If support vector points shift
+	- Plane shifts
+	- Hyperplane only depends on the support vectors
+		- Rest don't matter
+
+![](../../../img/svm-optimal-plane.png)
+
+# Linearly Separable
+- Not linearly separable
+![](../../../img/svm-non-linear.png)
+- Add another dimension
+	- $z=x^2+y^2$
+- Square of the distance of the point from the origin
+![](../../../img/svm-non-linear-project.png)
+- Now separable
+- Let $z=k$
+	- $k$ is a constant
+- Project linear separator back to 2D
+	- Get circle
+![](../../../img/svm-non-linear-separated.png)

AI/Classification/Supervised/Supervised.md

@@ -0,0 +1,23 @@
+
+# Gaussian Classifier
+- With $T$ labelled data
+$$q_t(i)=
+\begin{cases}
+    1 & \text{if class } i \\
+    0 & \text{otherwise}
+\end{cases}$$
+- Indicator function
+
+- Mean parameter
+$$\hat m_i=\frac{\sum_tq_t(i)o_t}{\sum_tq_t(i)}$$
+- Variance parameter
+$$\hat v_i=\frac{\sum_tq_t(i)(o_t-\hat m_i)^2}{\sum_tq_t(i)}$$
+
+- Distribution weight
+	- Class prior
+	- $P(N_i)$
+$$\hat c_i=\frac 1 T \sum_tq_t(i)$$
+
+$$\hat \mu_i=\frac{\sum_{t=1}^Tq_t(i)o_t}{\sum_{t=1}^Tq_t(i)}$$
+$$\hat\sum_i=\frac{\sum_{t=1}^Tq_t(i)(o_t-\mu_i)(o_t-\mu_i)^T}{\sum_{t=1}^Tq_t(i)}$$
+- For K-dimensional

AI/Learning.md

@@ -0,0 +1,63 @@
+# Supervised
+- Dataset with inputs manually annotated for desired output
+	- Desired output = supervisory signal
+	- Manually annotated = ground truth
+		- Annotated correct categories
+
+## Split data
+- Training set
+- Test set
+***Don't test on training data***
+
+## Top-K Accuracy
+- Whether correct answer appears in the top-k results
+
+## Confusion Matrix
+Samples described by ***feature vector***
+Dataset forms a matrix
+![](../img/confusion-matrix.png)
+
+# Un-Supervised
+- No example outputs given, learns how to categorise
+- No teacher or critic
+
+## Harder
+- Must identify relevant distinguishing features
+- Must decide on number of categories
+
+# Reinforcement Learning
+- No teacher - critic instead
+- Continued interaction with the environment
+- Minimise a scalar performance index
+
+![](../img/reinforcement-learning.png)
+
+- Critic
+	- Converts primary reinforcement to heuristic reinforcement
+	- Both scalar inputs
+- Delayed reinforcement
+	- System observes temporal sequence of stimuli
+	- Results in generation of heuristic reinforcement signal
+- Minimise cost-to-go function
+	- Expectation of cumulative cost of actions taken over sequence of steps
+	- Instead of just immediate cost
+	- Earlier actions may have been good
+		- Identify and feedback to environment
+- Closely related to dynamic programming
+
+## Difficulties
+- No teacher to provide desired response
+- Must solve temporal credit assignment problem
+	- Need to know which actions were the good ones
+
+# Fitting
+- Over-fitting
+	- Classifier too specific to training set
+	- Can't adequately generalise
+- Under-fitting
+	- Too general, not inferred enough detail
+	- Learns non-discriminative or non-desired pattern
+
+# ROC
+Receiver Operator Characteristic Curve
+![](../img/receiver-operator-curve.png)

AI/Neural Networks/Learning/Boltzmann.md

@@ -0,0 +1,30 @@
+- Stochastic
+- Recurrent structure
+- Binary operation (+/- 1)
+- Energy function
+
+$$E=-\frac 1 2 \sum_j\sum_k w_{kj}x_kx_j$$
+- $j\neq k$
+- No self-feedback
+- $x$ = neuron state
+- Neurons randomly flip from $x$ to $-x$
+
+$$P(x_k \rightarrow-x_k)=\frac 1 {1+e^{\frac{-\Delta E_k}{T}}}$$
+
+- Energy change based on pseudo-temperature
+	- System will reach thermal equilibrium
+- Delta E is the energy change resulting from the flip
+- Visible and hidden neurons
+	- Visible act as interface between network and environment
+	- Hidden always operate freely
+
+# Operation Modes
+- Clamped
+	- Visible neurons are clamped onto specific states determined by environment
+- Free-running
+	- All neurons able to operate freely
+- $\rho_{kj}^+$ = Correlation between states while clamped
+- $\rho_{kj}^-$ = Correlation between states while free
+- Both exist between +/- 1
+
+$$\Delta w_{kj}=\eta(\rho_{kj}^+-\rho_{kj}^-), \space j\neq k$$

AI/Neural Networks/Learning/Competitive Learning.md

@@ -0,0 +1,40 @@
+- Only single output neuron fires
+
+1. Set of homogeneous neurons with some randomly distributed synaptic weights
+	- Respond differently to given set of input patterns
+2. Limit imposed on strength of each neuron
+3. Mechanism to allow neurons to compete for right to respond to a given subset of inputs
+	- Only one output neuron active at a time
+	- Or only one neuron per group
+	- ***Winner-takes-all neuron***
+
+![](../../../img/comp-learning.png)
+
+- Lateral inhibition
+	- Neurons inhibit other neurons
+- Winning neuron must have highest induced local field for given input pattern
+	- Winning neuron is squashed to 1
+	- Others are clamped to 0
+
+$$y_k=
+\begin{cases}
+    1 & \text{if } v_k > v_j \text{ for all } j,j\neq k \\
+    0 & \text{otherwise}
+\end{cases}
+$$
+
+- Neuron has fixed amount of weight spread amongst input synapses
+	- Sums to 1
+- Learn by shifting weights from inactive to active input nodes
+	- Each input node relinquishes some proportion of weight
+	- Distributed amongst active nodes
+
+$$\Delta w_{kj}=
+\begin{cases}
+    \eta(x_j-w_{kj}) & \text{if neuron $k$ wins the competition}\\
+    0 & \text{if neuron $k$ loses the competition}
+\end{cases}$$
+
+- Individual neurons learn to specialise on ensembles of similar patterns
+	- Feature detectors
+![](../../../img/competitive-geometric.png)

AI/Neural Networks/Learning/Credit-Assignment Problem.md

@@ -0,0 +1,17 @@
+- Assigning credit/blame for outcomes to each internal decision
+- Loading Problem
+	- Loading a training set into the free parameters
+- Important to any learning machine attempting to improve performance in situations involving temporally extended behaviour
+
+Two Sub-problems:
+- ***Temporal*** credit-assignment problem
+	- Assigning credit for **outcomes** to **actions**
+	- Involves time when actions that deserve credit were taken
+	- Relevant when many actions taken and want to know which one was responsible
+- ***Structural*** credit-assignment problem
+	- Assigning credit for **actions** to **internal decisions**
+	- Involves internal structures of actions generated by system
+	- Relevant for identifying which component should have behaviour altered
+		- By how much
+
+- Important in MLPs when there are many hidden neurons

AI/Neural Networks/Learning/Hebbian.md

@@ -0,0 +1,55 @@
+*Time-dependent, highly local, strongly interactive*
+
+- Oldest learning algorithm
+- Increases synaptic efficiency as a function of the correlation between presynaptic and postsynaptic activities
+
+1. If two neurons on either side of a synapse are activated simultaneously/synchronously, then the strength of that synapse is selectively increased
+2. If two neurons on either side of a synapse are activated asynchronously, then that synapse is selectively weakened or eliminated
+
+- Hebbian synapse
+	- Time-dependent
+		- Depends on times of pre/post-synaptic signals
+	- Local
+	- Interactive
+		- Depends on both sides of synapse
+		- True interaction between pre/post-synaptic signals
+			- Cannot make prediction from either one by itself
+	- Conjunctional or correlational
+		- Based on conjunction of pre/post-synaptic signals
+		- Conjunctional synapse
+- Modification classifications
+	- Hebbian
+		- **Increases** strength with **positively** correlated pre/post-synaptic signals
+		- **Decreases** strength with **negatively** correlated pre/post-synaptic signals
+	- Anti-Hebbian
+		- **Decreases** strength with **positively** correlated pre/post-synaptic signals
+		- **Increases** strength with **negatively** correlated pre/post-synaptic signals
+		- Still Hebbian in nature, not in function
+	- Non-Hebbian
+		- Doesn't involve above correlations/time dependence etc
+
+# Mathematically
+$$\Delta w_{kj}(n)=F\left(y_k(n),x_j(n)\right)$$
+- Generally
+- All Hebbian
+
+![](../../../img/hebb-learning.png)
+
+## Hebb's Hypothesis
+$$\Delta w_{kj}(n)=\eta y_k(n)x_j(n)$$
+- Activity product rule
+- Exponential growth until saturation
+	- No information stored
+	- Selectivity lost
+
+## Covariance Hypothesis
+$$\Delta w_{kj}(n)=\eta(x_j-\bar x)(y_k-\bar y)$$
+- Characterised by perturbation from of pre/post-synaptic signals from their mean over a given time interval
+- Average $x$ and $y$ constitute thresholds
+- Intercept at y = y bar
+- Similar to learning in the hippocampus
+
+*Allows:*
+1. Convergence to non-trivial state
+	- When x = x bar or y = y bar
+2. Prediction of both synaptic potentiation and synaptic depression

AI/Neural Networks/Learning/Learning.md

@@ -0,0 +1,5 @@
+*Learning is a process by which the free parameters of a neural network are adapted through a process of stimulation by the environment in which the network is embedded. The type of learning is determined by the manner in which the parameter changes take place*
+
+1. The neural network is **stimulated** by an environment
+2. The network undergoes **changes in its free parameters** as a result of this stimulation
+3. The network **responds in a new way** to the environment as a result of the change in internal structure

AI/Neural Networks/Learning/README.md

@@ -0,0 +1 @@
+Learning.md

AI/Neural Networks/RNN/Autoencoder.md

@@ -0,0 +1,10 @@
+- Sequence of strokes for sketching
+	- LSTM backbone
+
+![](../../../img/rnn+autoencoder.png)
+
+# Variational
+- Learn mean and covariance to drive encoder stage
+	- Generate different outputs by sampling latent space
+
+![](../../../img/rnn+autoencoder-variational.png)

AI/Neural Networks/RNN/Deep Image Prior.md

@@ -0,0 +1,8 @@
+- Overfitted to image
+	- Learn weights necessary to reconstruct from white noise
+- Trained from scratch on single image
+	- Encodes prior for natural images
+	- De-noise images
+
+![](../../../img/deep-image-prior-arch.png)
+![](../../../img/deep-image-prior-results.png)

AI/Neural Networks/RNN/MoCo.md

@@ -0,0 +1,13 @@
+- Similar to SimCLR
+- Rich set of negatives
+	- Sampled from previous epochs in queue
+- Two function for pos/neg and anchor
+	- Pos/neg are delayed anchor weights
+		- Updated with momentum
+- Two delay mechanisms
+	- Two encoder functions
+	- Negative encoder queue
+
+![](../../../img/moco.png)
+
+$$\theta_k\leftarrow m\theta_k+(1-m)\theta_q$$

AI/Neural Networks/RNN/Representation Learning.md

@@ -0,0 +1,13 @@
+# Unsupervised
+
+- Auto-encoder FCN
+- Learns bottleneck (latent) representation
+	- Information rich
+	- $f(.)$ is CNN encoding function
+![](../../../img/unsup-representation-learning.png)
+
+# Supervised
+- Triplet loss
+	- Providing positive and negative requires supervision
+- Two losses
+![](../../../img/sup-representation-learning.png)

AI/Neural Networks/RNN/SimCLR.md

@@ -0,0 +1,10 @@
+1. Data augmentation
+	- Crop patches from images in batch
+	- Add colour jitter
+2. Within batch sample positive and negative
+	- Patches from same image are positive
+	- All other negative
+3. MLP layer to compute loss instead of bottleneck embedding
+	- Head network for function of bottleneck
+
+![](../../../img/simclr.png)