Solver360 Blog
Colorful deep dives into the math behind AI — linear algebra, calculus, probability, loss functions — plus the algorithms you can run in the matching Solver360 calculator.
Mathematics for AI
The math behind every solver
DBSCAN Explained: Density-Based Clustering, Epsilon, MinPts, and Outliers
How DBSCAN finds arbitrarily shaped clusters, labels noise, and why epsilon and minPts replace the k you must choose in k-means.
Feature Engineering Explained: Scaling, Encoding, and Polynomial Features
Standardization vs normalization, one-hot and label encoding, polynomial features, and the leakage mistakes that quietly inflate test scores.
Hyperparameter Tuning Explained: Grid Search, Random Search, and Bayesian Optimization
Parameters versus hyperparameters, why nested validation matters, and when grid search, random search, or Bayesian optimization is the better budget.
K-Fold Cross Validation Explained: Stratified Splits, Bias, and Model Selection
Why a single train/test split is noisy, how k-fold and stratified k-fold work, and how cross-validation should drive model and hyperparameter choices.
K-Nearest Neighbors Explained: Distance Metrics, Choosing K, and Classification
A practical k-NN guide covering Euclidean and Manhattan distance, majority vote, k selection, scaling, and the curse of dimensionality.
Logistic Regression Explained: Sigmoid Function, Odds, and Binary Classification
A complete logistic regression tutorial: logits, the sigmoid function, log-odds, cross-entropy loss, decision thresholds, and how to evaluate a binary classifier.
LSTM Explained: Forget Gate, Cell State, and Long-Term Memory in RNNs
Why vanilla RNNs forget, how LSTM gates protect a cell state, and when to use LSTM versus GRU for time series and sequence modeling.
Support Vector Machines Explained: Margins, Kernels, and Decision Boundaries
How SVMs find a maximum-margin hyperplane, why support vectors matter, and when linear, polynomial, and RBF kernels change the decision boundary.
Transformers Explained: Self-Attention, BERT, GPT, and Positional Encoding
Query, key, and value vectors, scaled dot-product attention, multi-head attention, and how BERT and GPT use the same transformer block differently.
XGBoost Explained: Gradient Boosting, Regularization, and Feature Importance
How XGBoost builds trees sequentially on residuals, why regularization and shrinkage matter, and how to read gain-based feature importance.
Bayes' Theorem for Machine Learning: Priors, Posteriors, and Naive Bayes
Priors, likelihoods, and posteriors with worked examples, plus the independence assumption, smoothing, and log-probabilities behind Naive Bayes.
Calculus for Machine Learning: Derivatives, Gradients, and the Chain Rule
How derivatives, partials, gradients, Jacobians, and the chain rule turn a loss into parameter updates — the calculus behind backpropagation.
Eigenvalues, SVD, and the Mathematics of PCA
Covariance, eigenvectors, singular values, and variance explained — the linear-algebra pipeline behind principal component analysis.
Information Theory for Deep Learning: Entropy, Cross-Entropy, and KL Divergence
Shannon entropy, cross-entropy loss, and KL divergence explained with the math that connects decision trees to neural network training.
Linear Algebra for Machine Learning: Vectors, Matrices, and Transformations
The core linear algebra used in AI: vectors, matrix multiplication, rank, projections, eigenvalues, and why neural networks are mostly matrix multiplies.
Loss Functions in Machine Learning: MSE, MAE, Likelihood, and Cross-Entropy
What a loss actually optimizes: MSE and MAE for regression, log loss and cross-entropy for classification, and how regularization adds extra terms.
Probability for Artificial Intelligence: Random Variables, Distributions, and Likelihood
A practical probability toolkit for AI: conditional probability, common distributions, likelihood, and maximum likelihood estimation.
Sigmoid, Softmax, and the Mathematics of Classification
Turn raw scores into probabilities: sigmoid, logits, softmax, temperature, log-sum-exp stability, and the last-layer math of classifiers.
Statistics for Machine Learning: Expectation, Variance, Bias, and Evaluation
The statistics that sit under model evaluation: sampling, bias-variance, correlation, confidence, and why train/test splits exist.
Vector Norms and Distance Metrics in AI: Euclidean, Manhattan, Cosine, and Beyond
L1, L2, cosine, and Minkowski distances — how the choice of metric changes k-NN, k-means, regularization, and nearest-neighbor geometry.
K-Means Clustering Explained: Centroids, Elbow Method, and When It Fails
A complete unsupervised-learning walkthrough of k-means: initialization, assignment and update steps, choosing k, silhouette scores, and density-based alternatives.
Decision Trees and Random Forests: Splits, Ensembles, and Feature Importance
From entropy and Gini impurity to bagging and out-of-bag error: how tree models partition space and why forests usually generalize better than a single deep tree.
Convolutional Neural Networks Guide: Filters, Pooling, and Image Recognition
A full tour of CNNs: convolution arithmetic, padding, stride, pooling, receptive fields, and how modern image models stack these blocks into classifiers.
Gradient Descent Complete Guide: Learning Rates, Variants, and Convergence
Understand why gradient descent works, how learning rate and batch size change the path, and when to use SGD, momentum, RMSProp, or Adam on real loss surfaces.
Neural Networks Explained from First Principles: Layers, Activations, and Backpropagation
Learn how multilayer perceptrons actually compute: weighted sums, nonlinear activations, forward pass, loss, and the backpropagation algorithm that trains them.
Linear Regression Explained: Math, Assumptions, Metrics, and Worked Examples
A complete practical guide to simple and multiple linear regression: ordinary least squares, residual analysis, R-squared, regularization, and how to interpret a fitted line with confidence.