Update notes

base/science-tech-maths/machine-learning/machine-learning.md

@@ -533,6 +533,8 @@ Many papers use the term long-tail learning to refer to class imbalance in multi
 
 ## Gradient descent
 
+The gradient measures the direction and the fastest rate of increase of a function $f$ at a given point.
+
 **Gradient always points in the direction of steepest ascent.**
 
 $$w^{(t+1)}=w^{(t)} - \eta\frac{\partial L}{\partial w}$$

base/science-tech-maths/machine-learning/metrics/metrics.md

@@ -15,8 +15,8 @@
 
 ## Precision-Recall curve vs ROC curve
 
-- Precision: how often the classifier is correct when it predicts positive $PRE = \frac{TP}{TP+FP}$
-- Recall: how often the classifier is correct for all positive instances
+- Precision: how often the classifier is correct when it predicts positive $PRE = \frac{TP}{TP+FP}$ "Of all the apples I picked from the basket, how many are actually good?"
+- Recall: how often the classifier is correct for all positive instances $REC = \frac{TP}{TP+FN}$ "Of all the good apples available, how many did I actually pick?"
 - [The Precision-Recall Plot Is More Informative than the ROC Plot When Evaluating Binary Classifiers on Imbalanced Datasets](https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0118432)
 - Indeed, ROC is useful when evaluating general-purpose classification, while AUPRC is the superior method when classifying rare events.
 - <https://towardsdatascience.com/why-you-should-stop-using-the-roc-curve-a46a9adc728>
@@ -30,7 +30,7 @@ Different threshold values will give different precision and recall values, and
 
 The AP is the area under the precision-recall curve.
 
-In object detection, the mAP is the mean of the average precision (AP) accross all classes.
+In object detection, the mAP is the mean of the average precision (AP) across all classes.
 
 ## MCC
 

base/science-tech-maths/maths/maths/maths.md

@@ -43,12 +43,13 @@ The second largest eigenvector is always orthogonal to the largest eigenvector,
 If all eigenvalues of A are:
 
 - positive: the matrix is positive definite.
-- positive or zero: positive semi-definite.
+- positive or zero: positive semi-definite.
 - negative: the matrix is negative definite.
 
 ## Derivatives
 
-- the gradient of $f$ is the vector containing all of the partial derivatives
+- The gradient of $f$ is the vector containing all of the partial derivatives.
+- The Jacobian is the matrix of all first-order partial derivatives of a vector-valued function (a function that takes a vector as an input and returns a vector as an output).
 - Hessian is the Jacobian of the gradient. Is symmetric.
 
 ### Finite differences