HMath.MachineLearning.Decomposition¶

Description

Dense decomposition and manifold reference algorithms. PCA signs and degenerate eigenspaces are not unique. t-SNE implements exact all-pairs Student-t KL gradients (van der Maaten & Hinton, JMLR 2008); UMAP uses the fuzzy union and full cross-entropy gradient (McInnes et al., arXiv:1802.03426) with caller-supplied positive a,b curve parameters. Neither is a Barnes-Hut, ANN, or stochastic negative-sampling implementation. Budgets are explicit.

On this page¶

API	Purpose
PCA	Fitted feature center,principal-component directions,and their explained variances; use transform functions for new observations.
pcaMean	Training feature means subtracted before projection.
pcaComponents	Principal component loading vectors in component order.
pcaVariances	Explained variances associated with the retained principal components.
fitPCA	Fit count principal axes from at least two observations; uses sample covariance (denominator n-1).
transformPCA	Center with the training mean and project onto retained axes; no whitening is applied.
inversePCA	Reconstruct original features from retained scores and restore the training mean.
IncrementalPCA	Exact sufficient statistics, including between-batch mean correction.
incrementalPCA	Create empty running covariance state for a positive feature dimension.
updatePCA	Merge observations by Welford updates.
mergePCA	Combine independent batch statistics with the between-mean scatter correction.
finishPCA	Extract principal axes from accumulated sample covariance; requires at least two observations.
KernelPCA	Centered training kernel state and normalized positive eigenvectors.
fitKernelPCA	Fit kernel principal axes; count must be smaller than n and within positive numerical rank.
transformKernelPCA	Center query kernels with training means before projection, without fitting query statistics.
truncatedSVD	Return reduced (U, singular values, V-transpose) from LAPACK, keeping the largest count values.
ICA	Training center and whitened unmixing directions; component signs/permutations are arbitrary.
fitICA	Symmetric FastICA with tanh nonlinearity (Hyvarinen 1999), whitening by sample covariance.
transformICA	Apply the fitted centering and unmixing directions to new observations.
FactorAnalysis	Fitted feature center,latent factor loadings,and diagonal feature-noise variances.
factorLoadings	Fitted feature-by-factor loading matrix.
factorNoise	Per-feature residual noise variances in the factor model.
fitFactorAnalysis	Gaussian factor-analysis EM with diagonal noise floor.
CCA	Paired training centers, canonical directions, and regularized canonical correlations.
fitCCA	Fit paired canonical directions with a strictly positive diagonal covariance ridge.
transformCCA	Project paired query rows with fitted training centers and canonical directions.
canonicalCorrelations	Singular values of the whitened cross-covariance, in descending order.
PLS	PLS2 regression state with predictor/response centers and a feature-by-response coefficient matrix.
fitPLS	PLS2 NIPALS with regression deflation; count cannot exceed the predictor rank.
predictPLS	Predict response rows using fitted PLS coefficients and training centers.
randomProjection	Seeded Gaussian projection with variance 1/dimension per coefficient.
classicalMDS	Classical metric MDS.
isomap	Build a symmetric k-neighbor graph, compute geodesics, then metric MDS.
tsne	Exact all-pairs t-SNE; perplexity must be strictly between one and n.
umap	Exact full fuzzy-set cross entropy, caller supplies fitted curve a,b.

Documentation

PCA

data PCA Source #

Fitted feature center,principal-component directions,and their explained variances; use transform functions for new observations.

Instances

Instances details

Show PCA Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods showsPrec :: Int -> PCA -> ShowS # show :: PCA -> String # showList :: [PCA] -> ShowS #
Eq PCA Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods (==) :: PCA -> PCA -> Bool # (/=) :: PCA -> PCA -> Bool #

pcaMean

pcaMean :: PCA -> [Double] Source #

Training feature means subtracted before projection.

pcaComponents

pcaComponents :: PCA -> Dataset Source #

Principal component loading vectors in component order.

pcaVariances

pcaVariances :: PCA -> [Double] Source #

Explained variances associated with the retained principal components.

fitPCA

fitPCA :: Int -> Dataset -> Either MathError PCA Source #

Fit count principal axes from at least two observations; uses sample covariance (denominator n-1).

transformPCA

transformPCA :: PCA -> Dataset -> Either MathError Dataset Source #

Center with the training mean and project onto retained axes; no whitening is applied.

inversePCA

inversePCA :: PCA -> Dataset -> Either MathError Dataset Source #

Reconstruct original features from retained scores and restore the training mean.

IncrementalPCA

data IncrementalPCA Source #

Exact sufficient statistics, including between-batch mean correction. Sample count, running mean, and centered scatter sufficient statistics.

Instances

Instances details

Show IncrementalPCA Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods showsPrec :: Int -> IncrementalPCA -> ShowS # show :: IncrementalPCA -> String # showList :: [IncrementalPCA] -> ShowS #
Eq IncrementalPCA Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods (==) :: IncrementalPCA -> IncrementalPCA -> Bool # (/=) :: IncrementalPCA -> IncrementalPCA -> Bool #

incrementalPCA

incrementalPCA :: Int -> Either MathError IncrementalPCA Source #

Create empty running covariance state for a positive feature dimension.

updatePCA

updatePCA :: IncrementalPCA -> Dataset -> Either MathError IncrementalPCA Source #

Merge observations by Welford updates. Empty batches preserve the previous state.

mergePCA

mergePCA :: IncrementalPCA -> IncrementalPCA -> Either MathError IncrementalPCA Source #

Combine independent batch statistics with the between-mean scatter correction.

finishPCA

finishPCA :: Int -> IncrementalPCA -> Either MathError PCA Source #

Extract principal axes from accumulated sample covariance; requires at least two observations.

KernelPCA

data KernelPCA Source #

Centered training kernel state and normalized positive eigenvectors.

Instances

Instances details

Show KernelPCA Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods showsPrec :: Int -> KernelPCA -> ShowS # show :: KernelPCA -> String # showList :: [KernelPCA] -> ShowS #
Eq KernelPCA Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods (==) :: KernelPCA -> KernelPCA -> Bool # (/=) :: KernelPCA -> KernelPCA -> Bool #

fitKernelPCA

fitKernelPCA :: Int -> Kernel -> Dataset -> Either MathError KernelPCA Source #

Fit kernel principal axes; count must be smaller than n and within positive numerical rank.

transformKernelPCA

transformKernelPCA :: KernelPCA -> Dataset -> Either MathError Dataset Source #

Center query kernels with training means before projection, without fitting query statistics.

truncatedSVD

truncatedSVD :: Int -> Dataset -> Either MathError (Dataset, [Double], Dataset) Source #

Return reduced (U, singular values, V-transpose) from LAPACK, keeping the largest count values.

ICA

data ICA Source #

Training center and whitened unmixing directions; component signs/permutations are arbitrary.

Instances

Instances details

Show ICA Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods showsPrec :: Int -> ICA -> ShowS # show :: ICA -> String # showList :: [ICA] -> ShowS #
Eq ICA Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods (==) :: ICA -> ICA -> Bool # (/=) :: ICA -> ICA -> Bool #

fitICA

fitICA :: MLConfig -> Int -> Dataset -> Either MathError (Fit ICA) Source #

Symmetric FastICA with tanh nonlinearity (Hyvarinen 1999), whitening by sample covariance. Convergence accounts for independent component signs.

transformICA

transformICA :: ICA -> Dataset -> Either MathError Dataset Source #

Apply the fitted centering and unmixing directions to new observations.

FactorAnalysis

data FactorAnalysis Source #

Fitted feature center,latent factor loadings,and diagonal feature-noise variances.

Instances

Instances details

Show FactorAnalysis Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods showsPrec :: Int -> FactorAnalysis -> ShowS # show :: FactorAnalysis -> String # showList :: [FactorAnalysis] -> ShowS #
Eq FactorAnalysis Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods (==) :: FactorAnalysis -> FactorAnalysis -> Bool # (/=) :: FactorAnalysis -> FactorAnalysis -> Bool #

factorLoadings

factorLoadings :: FactorAnalysis -> Dataset Source #

Fitted feature-by-factor loading matrix.

factorNoise

factorNoise :: FactorAnalysis -> [Double] Source #

Per-feature residual noise variances in the factor model.

fitFactorAnalysis

fitFactorAnalysis :: MLConfig -> Int -> Double -> Dataset -> Either MathError (Fit FactorAnalysis) Source #

Gaussian factor-analysis EM with diagonal noise floor. Unlike PCA, each observed feature has its own learned residual variance.

CCA

data CCA Source #

Paired training centers, canonical directions, and regularized canonical correlations.

Instances

Instances details

Show CCA Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods showsPrec :: Int -> CCA -> ShowS # show :: CCA -> String # showList :: [CCA] -> ShowS #
Eq CCA Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods (==) :: CCA -> CCA -> Bool # (/=) :: CCA -> CCA -> Bool #

fitCCA

fitCCA :: Int -> Double -> Dataset -> Dataset -> Either MathError CCA Source #

Fit paired canonical directions with a strictly positive diagonal covariance ridge.

transformCCA

transformCCA :: CCA -> Dataset -> Dataset -> Either MathError (Dataset, Dataset) Source #

Project paired query rows with fitted training centers and canonical directions.

canonicalCorrelations

canonicalCorrelations :: CCA -> [Double] Source #

Singular values of the whitened cross-covariance, in descending order.

PLS

data PLS Source #

PLS2 regression state with predictor/response centers and a feature-by-response coefficient matrix.

Instances

Instances details

Show PLS Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods showsPrec :: Int -> PLS -> ShowS # show :: PLS -> String # showList :: [PLS] -> ShowS #
Eq PLS Source #
Instance details Defined in HMath.MachineLearning.Decomposition Methods (==) :: PLS -> PLS -> Bool # (/=) :: PLS -> PLS -> Bool #

fitPLS

fitPLS :: MLConfig -> Int -> Dataset -> Dataset -> Either MathError (Fit PLS) Source #

PLS2 NIPALS with regression deflation; count cannot exceed the predictor rank. Inner fixed-point iteration obeys the supplied convergence budget.

predictPLS

predictPLS :: PLS -> Dataset -> Either MathError Dataset Source #

Predict response rows using fitted PLS coefficients and training centers.

randomProjection

randomProjection :: MLConfig -> Int -> Dataset -> Either MathError Dataset Source #

Seeded Gaussian projection with variance 1/dimension per coefficient. Returns transformed rows.

classicalMDS

classicalMDS :: Int -> Dataset -> Either MathError Dataset Source #

Classical metric MDS. Negative eigenvalues are omitted; requesting a dimension beyond positive rank fails instead of creating imaginary axes.

isomap

isomap :: Int -> Int -> Dataset -> Either MathError Dataset Source #

Build a symmetric k-neighbor graph, compute geodesics, then metric MDS. Disconnected graphs fail.

tsne

tsne :: MLConfig -> Int -> Double -> Dataset -> Either MathError (Fit Dataset) Source #

Exact all-pairs t-SNE; perplexity must be strictly between one and n.

umap

umap :: MLConfig -> Int -> Int -> Double -> Double -> Dataset -> Either MathError (Fit Dataset) Source #

Exact full fuzzy-set cross entropy, caller supplies fitted curve a,b. Local connectivity rho is the first positive neighbor distance. This dense reference resolves every repulsive pair instead of sampling negatives.