spectral-clustering (empty) → 0.2.1.1
raw patch · 7 files changed
+1363/−0 lines, 7 filesdep +basedep +clusteringdep +eigensetup-changed
Dependencies added: base, clustering, eigen, hmatrix, hmatrix-svdlibc, mwc-random, safe, sparse-linear-algebra, statistics, vector
Files
- LICENSE +674/−0
- Setup.hs +2/−0
- spectral-clustering.cabal +37/−0
- src/Math/Clustering/Spectral/Dense.hs +94/−0
- src/Math/Clustering/Spectral/Eigen/AdjacencyMatrix.hs +181/−0
- src/Math/Clustering/Spectral/Eigen/FeatureMatrix.hs +160/−0
- src/Math/Clustering/Spectral/Sparse.hs +215/−0
+ LICENSE view
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+ Setup.hs view
@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain
+ spectral-clustering.cabal view
@@ -0,0 +1,37 @@+name: spectral-clustering+version: 0.2.1.1+synopsis: Library for spectral clustering.+description: Spectral clustering of a matrix.+homepage: http://github.com/GregorySchwartz/spectral-clustering#readme+license: GPL-3+license-file: LICENSE+author: Gregory W. Schwartz+maintainer: gsch@mail.med.upenn.edu+copyright: 2018 Gregory W. Schwartz+category: Math+build-type: Simple+-- extra-source-files:+cabal-version: >=1.10++library+ hs-source-dirs: src+ exposed-modules: Math.Clustering.Spectral.Dense+ , Math.Clustering.Spectral.Sparse+ , Math.Clustering.Spectral.Eigen.FeatureMatrix+ , Math.Clustering.Spectral.Eigen.AdjacencyMatrix+ build-depends: base >= 4.7 && < 5+ , clustering+ , eigen+ , hmatrix+ , hmatrix-svdlibc+ , mwc-random+ , safe+ , sparse-linear-algebra+ , statistics+ , vector+ ghc-options: -O2+ default-language: Haskell2010++source-repository head+ type: git+ location: https://github.com/GregorySchwartz/spectral-clustering
+ src/Math/Clustering/Spectral/Dense.hs view
@@ -0,0 +1,94 @@+{- Math.Clustering.Spectral.Dense+Gregory W. Schwartz++Collects the functions pertaining to spectral clustering.+-}++module Math.Clustering.Spectral.Dense+ ( spectralClusterKNorm+ , spectralClusterNorm+ , spectralNorm+ , getDegreeMatrix+ , AdjacencyMatrix (..)+ ) where++-- Remote+import Data.Bool (bool)+import Data.Function (on)+import Data.List (sortBy)+import Data.Maybe (fromMaybe)+import Safe (headMay)+import qualified AI.Clustering.KMeans as K+import qualified Data.Vector as V+import qualified Data.Vector.Unboxed as U+import qualified Numeric.LinearAlgebra as H+import qualified Statistics.Quantile as S++-- Local++type LabelVector = H.Vector Double+type AdjacencyMatrix = H.Matrix Double++-- | Returns the clustering of eigenvectors with the second smallest eigenvalues+-- and on of the symmetric normalized Laplacian L. Computes real symmetric part+-- of L, so ensure the input is real and symmetric. Diagonal should be 0s for+-- adjacency matrix. Clusters the eigenvector using kmeans into k groups from e+-- eigenvectors.+spectralClusterKNorm :: Int -> Int -> AdjacencyMatrix -> LabelVector+spectralClusterKNorm e k mat+ | H.rows mat < 1 = H.fromList []+ | H.rows mat == 1 = H.fromList [0]+ | otherwise = V.convert+ . U.map fromIntegral+ . K.membership+ . (\x -> K.kmeansBy k x id K.defaultKMeansOpts)+ . V.fromList+ . fmap V.convert+ . H.toRows+ . H.fromColumns+ . fmap H.normalize -- Normalize within eigenvectors (columns).+ . H.toColumns+ . H.fromRows+ . spectralNorm 1 e+ $ mat++-- | Returns the eigenvector with the second smallest eigenvalue of the+-- symmetric normalized Laplacian L. Computes real symmetric part of L, so+-- ensure the input is real and symmetric. Diagonal should be 0s for adjacency+-- matrix.+spectralClusterNorm :: AdjacencyMatrix -> LabelVector+spectralClusterNorm mat+ | H.rows mat < 1 = H.fromList []+ | H.rows mat == 1 = H.fromList [0]+ | otherwise =+ H.cmap (bool 0 1 . (>= 0)) . mconcat . spectralNorm 2 1 $ mat++-- | Returns the eigenvectors with the Nth smallest eigenvalue and on of the+-- symmetric normalized Laplacian L. Computes real symmetric part of L, so+-- ensure the input is real and symmetric. Diagonal should be 0s for adjacency+-- matrix.+spectralNorm :: Int -> Int -> AdjacencyMatrix -> [H.Vector Double]+spectralNorm n e mat+ | e < 1 = error "Less than 1 eigenvector chosen for clustering."+ | n < 1 = error "N < 1, cannot go before first eigenvector."+ | otherwise = H.toRows+ . flip (H.??) (H.All, H.TakeLast e)+ . flip (H.??) (H.All, H.DropLast (n - 1))+ . snd+ . H.eigSH+ $ lNorm+ where+ lNorm = H.sym $ i - mconcat [invD, mat, invD]+ invD = H.diag+ . H.cmap (\x -> if x == 0 then x else x ** (- 1 / 2))+ . getDegreeVector+ $ mat+ i = H.ident . H.rows $ mat++-- | Obtain the degree matrix.+getDegreeMatrix :: AdjacencyMatrix -> H.Matrix Double+getDegreeMatrix = H.diag . getDegreeVector++-- | Obtain the degree vector.+getDegreeVector :: AdjacencyMatrix -> H.Vector Double+getDegreeVector = H.vector . fmap H.sumElements . H.toRows
+ src/Math/Clustering/Spectral/Eigen/AdjacencyMatrix.hs view
@@ -0,0 +1,181 @@+{- Math.Clustering.Spectral.Eigen.AdjacencyMatrix+Gregory W. Schwartz++Collects the functions pertaining to spectral clustering.+-}++{-# LANGUAGE BangPatterns #-}++module Math.Clustering.Spectral.Eigen.AdjacencyMatrix+ ( spectralClusterKNorm+ , spectralClusterNorm+ , spectralNorm+ , getDegreeMatrix+ , secondLeft+ , AdjacencyMatrix (..)+ ) where++-- Remote+import Control.Monad (replicateM)+import Data.Bool (bool)+import Data.Function (on)+import Data.List (sortBy)+import Data.Maybe (fromMaybe)+import Safe (headMay)+import System.Random.MWC (createSystemRandom, uniform)+import qualified AI.Clustering.KMeans as K+import qualified Data.Eigen.SparseMatrix as S+import qualified Data.Vector as V+import qualified Data.Vector.Unboxed as U+import qualified Numeric.LinearAlgebra as H+import qualified Numeric.LinearAlgebra.Devel as H+import qualified Numeric.LinearAlgebra.SVD.SVDLIBC as SVD+import qualified Statistics.Quantile as Stat++-- Local++type LabelVector = S.SparseMatrixXd+type AdjacencyMatrix = S.SparseMatrixXd++-- | Returns the clustering of the eigenvectors with the second smallest+-- eigenvalues of the symmetric normalized Laplacian L. Computes real symmetric+-- part of L, so ensure the input is real and symmetric. Diagonal should be 0s+-- for adjacency matrix. Clusters the eigenvector using kmeans into k groups.+spectralClusterKNorm :: Int -> Int -> AdjacencyMatrix -> LabelVector+spectralClusterKNorm e k mat+ | S.rows mat < 1 = S.fromDenseList [[]]+ | S.rows mat == 1 = S.fromDenseList [[0]]+ | otherwise = kmeansVec k . spectralNorm 1 e $ mat++-- | Returns the clustering of the eigenvectors with the second smallest+-- eigenvalues of the symmetric normalized Laplacian L. Computes real symmetric+-- part of L, so ensure the input is real and symmetric. Diagonal should be 0s+-- for adjacency matrix. Clusters the eigenvector by sign.+spectralClusterNorm :: AdjacencyMatrix -> LabelVector+spectralClusterNorm mat+ | S.rows mat < 1 = S.fromDenseList [[]]+ | S.rows mat == 1 = S.fromDenseList [[0]]+ | otherwise = S.fromDenseList+ . (fmap . fmap) (bool 0 1 . (>= 0))+ . S.toDenseList+ . spectralNorm 2 1+ $ mat++-- | Returns the eigenvector with the second smallest eigenvalue (or N start)+-- and E on of the symmetric normalized Laplacian L. Computes real symmetric+-- part of L, so ensure the input is real and symmetric. Diagonal should be 0s+-- for adjacency matrix. Uses I + Lnorm instead of I - Lnorm to find second+-- largest singular value instead of second smallest for Lnorm.+spectralNorm :: Int -> Int -> AdjacencyMatrix -> S.SparseMatrixXd+spectralNorm n e mat+ | e < 1 = error "Less than 1 eigenvector chosen for clustering."+ | n < 1 = error "N < 1, cannot go before first eigenvector."+ | otherwise = secondLeft n e lNorm+ where+ lNorm = i + (S.transpose invRootD * (mat * invRootD))+ invRootD = S.diagRow 0+ . S._map (\x -> if x == 0 then x else x ** (- 1 / 2))+ . getDegreeVector+ $ mat+ i = S.ident . S.rows $ mat++-- | Second largest eigenvector. Unused, untested, don't use.+secondLargest :: S.SparseMatrixXd -> IO S.SparseMatrixXd+secondLargest a = do+ (first, firstVal) <- powerIt a++ let firstScale = S.scale firstVal (first * S.transpose first)+ b = a - firstScale++ fmap fst $ powerIt b++-- | Rayleigh quotient. Takes a matrix and an eigenvector to find the+-- corresponding eigenvalue. Unused, untested, don't use.+rayQuot :: S.SparseMatrixXd -> S.SparseMatrixXd -> Double+rayQuot a x = ((S.transpose (a * x) * x) S.! (0, 0))+ / ((S.transpose x * x) S.! (0, 0))++-- | Power iteration. Unused, untested, don't use.+powerIt :: S.SparseMatrixXd -> IO (S.SparseMatrixXd, Double)+powerIt a = do+ g <- createSystemRandom+ start <-+ fmap (S.fromDenseList . fmap (: [])) . replicateM (S.rows a) . uniform $ g+ let go+ :: Int+ -> Double+ -> S.SparseMatrixXd -- Eigenvector guess.+ -> Double -- Eigenvalue guess.+ -> (S.SparseMatrixXd, Double)+ go !i !e !b !lambda =+ if (abs (lambda' - lambda) < e) || (i < 0)+ then (b', lambda')+ else go (i - 1) e b' lambda'+ where+ absMat :: S.SparseMatrixXd -> S.SparseMatrixXd+ absMat = S._map abs+ b' :: S.SparseMatrixXd+ b' = S.scale (1 / maxElement ab) ab+ ab :: S.SparseMatrixXd+ ab = a * b+ maxElement = maximum . fmap (\(_, _, !x) -> abs x) . S.toList+ lambda' = S.norm ab / S.norm b+ return $ go 1000 0.000001 start (1 / 0)++-- | Executes kmeans to cluster a one dimensional vector.+kmeansVec :: Int -> S.SparseMatrixXd -> LabelVector+kmeansVec k = S.fromDenseList+ . fmap ((:[]) . fromIntegral)+ . U.toList+ . K.membership+ . (\x -> K.kmeansBy k x id K.defaultKMeansOpts)+ . V.fromList+ . fmap U.fromList+ . concatMap S.toDenseList+ . S.getRows+ . S.fromCols+ . fmap normNormalize+ . S.getCols++-- | Normalize by the norm of a vector.+normNormalize :: S.SparseMatrixXd -> S.SparseMatrixXd+normNormalize xs = S._map (/ norm) xs+ where+ norm = S.norm xs++-- | Obtain the second largest value singular vector (or Nth) and E on of a+-- sparse matrix.+secondLeft :: Int -> Int -> S.SparseMatrixXd -> S.SparseMatrixXd+secondLeft n e m = S.transpose+ . S.fromDenseList+ . fmap H.toList+ . drop (n - 1)+ . H.toRows+ . (\(!x, _, _) -> x)+ . SVD.sparseSvd (e + (n - 1))+ . H.mkCSR+ . fmap (\(!i, !j, !x) -> ((i, j), x))+ . S.toList+ $ m++-- | Obtain the second largest value singular vector of a sparse matrix.+denseSecondLeft :: S.SparseMatrixXd -> S.SparseMatrixXd+denseSecondLeft m = S.fromDenseList+ . fmap (:[])+ . H.toList+ . (!! 2)+ . H.toColumns+ . (\(!x, _, _) -> x)+ . H.svd+ . H.assoc (S.rows m, S.cols m) 0+ . fmap (\(!i, !j, !x) -> ((i, j), x))+ . S.toList+ $ m++-- | Obtain the degree matrix. Faster for columns.+getDegreeMatrix :: AdjacencyMatrix -> S.SparseMatrixXd+getDegreeMatrix = S.diagRow 0 . getDegreeVector++-- | Obtain the degree vector. Faster for columns.+getDegreeVector :: AdjacencyMatrix -> S.SparseMatrixXd+getDegreeVector = S.getColSums
+ src/Math/Clustering/Spectral/Eigen/FeatureMatrix.hs view
@@ -0,0 +1,160 @@+{- Math.Clustering.Spectral.Eigen.FeatureMatrix+Gregory W. Schwartz++Collects the functions pertaining to sparse spectral clustering.+-}++{-# LANGUAGE BangPatterns #-}++module Math.Clustering.Spectral.Eigen.FeatureMatrix+ ( B (..)+ , B1 (..)+ , B2 (..)+ , spectral+ , spectralCluster+ , spectralClusterK+ , getB+ , b1ToB2+ , getSimilarityFromB2+ ) where++-- Remote+import Data.Bool (bool)+import Data.Function (on)+import Data.List (sortBy)+import Data.Maybe (fromMaybe)+import qualified AI.Clustering.KMeans as K+import qualified Data.Eigen.SparseMatrix as S+import qualified Data.Vector as V+import qualified Data.Vector.Storable as VS+import qualified Data.Vector.Unboxed as U+import qualified Numeric.LinearAlgebra as H+import qualified Numeric.LinearAlgebra.Devel as H+import qualified Numeric.LinearAlgebra.SVD.SVDLIBC as SVD+import Debug.Trace++-- Local++type LabelVector = S.SparseMatrixXd+newtype B1 = B1 { unB1 :: S.SparseMatrixXd } deriving (Show)+newtype B2 = B2 { unB2 :: S.SparseMatrixXd } deriving (Show)+newtype D = D { unD :: S.SparseMatrixXd } deriving (Show)+newtype C = C { unC :: S.SparseMatrixXd } deriving (Show)+newtype B = B { unB :: S.SparseMatrixXd } deriving (Show)++-- | Normalize the input matrix by column. Here, columns are features.+b1ToB2 :: B1 -> B2+b1ToB2 (B1 b1) =+ B2+ . S._imap (\ i j x+ -> (log (fromIntegral n / (fromMaybe 0 $ dVec VS.!? j))) * x+ )+ $ b1+ where+ dVec :: VS.Vector Double+ dVec = maybe (error "Cannot get number of non-zeros.") (VS.map fromIntegral)+ . S.innerNNZs+ . S.uncompress+ $ b1+ n = S.rows b1++-- | Euclidean norm each row.+b2ToB :: B2 -> B+b2ToB (B2 b2) =+ B+ . S._imap (\ i j x+ -> x / (fromMaybe (error "Norm is 0.") $ eVec VS.!? i)+ )+ $ b2+ where+ eVec :: VS.Vector Double+ eVec = VS.fromList . fmap S.norm . S.getRows $ b2++-- | Get the diagonal transformed B matrix.+bToD :: B -> D+bToD (B b) = D . S.diagCol 0 $ b * ((S.transpose b) * S.ones n)+ where+ n = S.rows b++-- | Get the matrix C as input for SVD.+bdToC :: B -> D -> C+bdToC (B b) (D d) = C $ (S._map (\x -> x ** (- 1 / 2)) d) * b++-- | Obtain the second largest value singular vector (or Nth) and E on of a+-- sparse matrix.+secondLeft :: Int -> Int -> S.SparseMatrixXd -> S.SparseMatrixXd+secondLeft n e m = S.transpose+ . S.fromDenseList+ . fmap H.toList+ . drop (n - 1)+ . H.toRows+ . (\(!x, _, _) -> x)+ . SVD.sparseSvd (e + (n - 1))+ . H.mkCSR+ . fmap (\(!i, !j, !x) -> ((i, j), x))+ . S.toList+ $ m++-- | Get the normalized matrix B from an input matrix where the features are+-- columns and rows are observations. Optionally, do not normalize.+getB :: Bool -> S.SparseMatrixXd -> B+getB True = b2ToB . b1ToB2 . B1+getB False = b2ToB . B2++-- | Returns the second left singular vector (or Nth) of a sparse spectral+-- process. Assumes the columns are features and rows are observations. B is the+-- normalized matrix (from getB). See Shu et al., "Efficient Spectral+-- Neighborhood Blocking for Entity Resolution", 2011.+spectral :: Int -> Int -> B -> S.SparseMatrixXd+spectral n e b = secondLeft n e . unC . bdToC b . bToD $ b++-- | Returns a vector of cluster labels for two groups by finding the second+-- left singular vector of a special normalized matrix. Assumes the columns are+-- features and rows are observations. B is the normalized matrix (from getB).+-- See Shu et al., "Efficient Spectral Neighborhood Blocking for Entity+-- Resolution", 2011.+spectralCluster :: B -> LabelVector+spectralCluster (B b)+ | S.rows b < 1 = S.fromDenseList [[]]+ | S.rows b == 1 = S.fromDenseList [[0]]+ | otherwise = S.fromDenseList+ . (fmap . fmap) (bool 0 1 . (>= 0))+ . S.toDenseList+ . spectral 2 1+ $ B b++-- | Returns a vector of cluster labels for two groups by finding the largest+-- singular vectors and on of a special normalized matrix and running kmeans.+-- Assumes the columns are features and rows are observations. B is the+-- normalized matrix (from getB). See Shu et al., "Efficient Spectral+-- Neighborhood Blocking for Entity Resolution", 2011.+spectralClusterK :: Int -> Int -> B -> LabelVector+spectralClusterK e k (B b)+ | S.rows b < 1 = S.fromDenseList [[]]+ | S.rows b == 1 = S.fromDenseList [[0]]+ | otherwise = S.fromDenseList+ . fmap ((:[]) . fromIntegral)+ . U.toList+ . K.membership+ . (\x -> K.kmeansBy k x id K.defaultKMeansOpts)+ . V.fromList+ . fmap U.fromList+ . concatMap S.toDenseList+ . S.getRows+ . S.fromCols+ . fmap normNormalize+ . S.getCols+ . spectral 1 e+ $ B b++-- | Normalize by the norm of a vector.+normNormalize :: S.SparseMatrixXd -> S.SparseMatrixXd+normNormalize xs = S._map (/ norm) xs+ where+ norm = S.norm xs++-- | Get the cosine similarity between two rows using B2.+getSimilarityFromB2 :: B2 -> Int -> Int -> Double+getSimilarityFromB2 (B2 b2) i j =+ (((S.getRow i b2) * (S.transpose $ S.getRow j b2)) S.! (0, 0))+ / (S.norm (S.getRow i b2) * S.norm (S.getRow j b2))
+ src/Math/Clustering/Spectral/Sparse.hs view
@@ -0,0 +1,215 @@+{- Math.Clustering.Spectral.Sparse+Gregory W. Schwartz++Collects the functions pertaining to sparse spectral clustering.+-}++{-# LANGUAGE BangPatterns #-}++module Math.Clustering.Spectral.Sparse+ ( B (..)+ , B1 (..)+ , B2 (..)+ , AdjacencyMatrix (..)+ , spectral+ , spectralCluster+ , spectralClusterK+ , spectralNorm+ , spectralClusterNorm+ , spectralClusterKNorm+ , getB+ , b1ToB2+ , getSimilarityFromB2+ ) where++-- Remote+import Data.Bool (bool)+import Data.Maybe (fromMaybe)+import Data.Function (on)+import Data.List (sortBy, foldl1')+import Safe (headMay)+import qualified AI.Clustering.KMeans as K+import qualified Data.Sparse.Common as S+import qualified Data.Vector as V+import qualified Data.Vector.Unboxed as U+import qualified Numeric.LinearAlgebra as H+import qualified Numeric.LinearAlgebra.Devel as H+import qualified Numeric.LinearAlgebra.SVD.SVDLIBC as SVD+import Debug.Trace++-- Local++type LabelVector = S.SpVector Double+type AdjacencyMatrix = S.SpMatrix Double+newtype B1 = B1 { unB1 :: S.SpMatrix Double } deriving (Show)+newtype B2 = B2 { unB2 :: S.SpMatrix Double } deriving (Show)+newtype D = D { unD :: S.SpMatrix Double } deriving (Show)+newtype C = C { unC :: S.SpMatrix Double } deriving (Show)+newtype B = B { unB :: S.SpMatrix Double } deriving (Show)++-- | Normalize the input matrix by column. Here, columns are features.+b1ToB2 :: B1 -> B2+b1ToB2 (B1 b1) =+ B2+ . S.fromListSM (n, m)+ . fmap (\ (!i, !j, !x)+ -> (i, j, (log (fromIntegral n / (S.lookupDenseSV j dVec))) * x)+ )+ . S.toListSM+ $ b1+ where+ dVec :: S.SpVector Double+ dVec = S.vr+ . fmap (sum . fmap (\x -> if x > 0 then 1 else 0))+ . S.toRowsL -- faster than toColsL.+ . S.transposeSM+ $ b1+ n = S.nrows b1+ m = S.ncols b1++-- | Euclidean norm each row.+b2ToB :: B2 -> B+b2ToB (B2 b2) =+ B+ . S.fromListSM (n, m)+ . fmap (\(!i, !j, !x) -> (i, j, x / (S.lookupDenseSV i eVec)))+ . S.toListSM+ $ b2+ where+ eVec :: S.SpVector Double+ eVec = S.vr . fmap S.norm2 . S.toRowsL $ b2+ n = S.nrows b2+ m = S.ncols b2++-- | Find the Euclidean norm of a vector.+norm2 :: S.SpVector Double -> Double+norm2 = sqrt . sum . fmap (** 2)++-- | Get the diagonal transformed B matrix.+bToD :: B -> D+bToD (B b) = D+ . S.diagonalSM+ . flip S.extractCol 0+ $ b+ S.#~# ((S.transposeSM b) S.#~# (S.fromColsL [S.onesSV n]))+ where+ n = S.nrows b++-- | Get the matrix C as input for SVD.+bdToC :: B -> D -> C+bdToC (B b) (D d) = C $ (fmap (\x -> x ** (- 1 / 2)) d) S.#~# b++-- | Obtain the second left singular vector (or N earlier) and E on of a sparse+-- matrix.+secondLeft :: Int -> Int -> S.SpMatrix Double -> [S.SpVector Double]+secondLeft n e m =+ fmap (S.sparsifySV . S.fromListDenseSV e . drop (n - 1) . H.toList)+ . H.toColumns+ . (\(!x, _, _) -> x)+ . SVD.sparseSvd (e + (n - 1))+ . H.mkCSR+ . fmap (\(!i, !j, !x) -> ((i, j), x))+ . S.toListSM+ $ m++-- | Get the normalized matrix B from an input matrix where the features are+-- columns and rows are observations. Optionally, do not normalize.+getB :: Bool -> S.SpMatrix Double -> B+getB True = b2ToB . b1ToB2 . B1+getB False = b2ToB . B2++-- | Returns the second left singular vector (or from N) and E on of a sparse+-- spectral process. Assumes the columns are features and rows are observations.+-- B is the normalized matrix (from getB). See Shu et al., "Efficient Spectral+-- Neighborhood Blocking for Entity Resolution", 2011.+spectral :: Int -> Int -> B -> [S.SpVector Double]+spectral n e b+ | e < 1 = error "Less than 1 eigenvector chosen for clustering."+ | n < 1 = error "N < 1, cannot go before first eigenvector."+ | otherwise = secondLeft n e . unC . bdToC b . bToD $ b++-- | Returns a vector of cluster labels for two groups by finding the second+-- left singular vector of a special normalized matrix. Assumes the columns are+-- features and rows are observations. B is the normalized matrix (from getB).+-- See Shu et al., "Efficient Spectral Neighborhood Blocking for Entity+-- Resolution", 2011.+spectralCluster :: B -> LabelVector+spectralCluster (B b)+ | S.nrows b < 1 = S.zeroSV 0+ | S.nrows b == 1 = S.zeroSV 1+ | otherwise = S.sparsifySV+ . S.vr+ . fmap (bool 0 1 . (>= 0))+ . S.toDenseListSV+ . foldl1' S.concatSV+ . spectral 2 1+ $ B b++-- | Returns a vector of cluster labels for two groups by finding the second+-- left singular vector and on of a special normalized matrix and running kmeans.+-- Assumes the columns are features and rows are observations. B is the+-- normalized matrix (from getB). See Shu et al., "Efficient Spectral+-- Neighborhood Blocking for Entity Resolution", 2011.+spectralClusterK :: Int -> Int -> B -> LabelVector+spectralClusterK e k (B b)+ | S.nrows b < 1 = S.zeroSV 0+ | S.nrows b == 1 = S.zeroSV 1+ | otherwise = kmeansVec k . spectral 1 e $ B b++-- | Executes kmeans to cluster a vector.+kmeansVec :: Int -> [S.SpVector Double] -> LabelVector+kmeansVec k = S.sparsifySV+ . S.vr+ . fmap fromIntegral+ . U.toList+ . K.membership+ . (\x -> K.kmeansBy k x id K.defaultKMeansOpts)+ . V.fromList+ . fmap (U.fromList . S.toDenseListSV)+ . S.toRowsL+ . S.fromColsL+ . fmap S.normalize2+ . S.toColsL+ . S.transpose+ . S.fromColsL++-- | Get the cosine similarity between two rows using B2.+getSimilarityFromB2 :: B2 -> Int -> Int -> Double+getSimilarityFromB2 (B2 b2) i j =+ S.dot (S.extractRow b2 i) (S.extractRow b2 j)+ / (S.norm2 (S.extractRow b2 i) * S.norm2 (S.extractRow b2 j))++-- | Returns the eigenvector with the second smallest eigenvalue (or N start)+-- and E on of the symmetric normalized Laplacian L. Computes real symmetric+-- part of L, so ensure the input is real and symmetric. Diagonal should be 0s+-- for adjacency matrix. Uses I + Lnorm instead of I - Lnorm to find second+-- largest singular value instead of second smallest for Lnorm.+spectralNorm :: Int -> Int -> AdjacencyMatrix -> [S.SpVector Double]+spectralNorm n e mat = secondLeft n e lNorm+ where+ lNorm = i S.^+^ (S.transpose invRootD S.#~# (mat S.#~# invRootD))+ invRootD = S.diagonalSM+ . S.vr+ . fmap ((\x -> if x == 0 then x else x ** (- 1 / 2)) . sum)+ . S.toRowsL+ $ mat+ i = S.eye . S.nrows $ mat++-- | Returns the eigenvector with the second smallest eigenvalue and on of the+-- symmetric normalized Laplacian L. Computes real symmetric part of L, so+-- ensure the input is real and symmetric. Diagonal should be 0s for adjacency+-- matrix. Clusters the eigenvector using kmeans into k groups.+spectralClusterKNorm :: Int -> Int -> AdjacencyMatrix -> LabelVector+spectralClusterKNorm e k = kmeansVec k . spectralNorm 1 e++-- | Returns the eigenvector with the second smallest eigenvalue of the+-- symmetric normalized Laplacian L. Computes real symmetric part of L, so+-- ensure the input is real and symmetric. Diagonal should be 0s for adjacency+-- matrix. Clusters the eigenvector by sign.+spectralClusterNorm :: AdjacencyMatrix -> LabelVector+spectralClusterNorm = S.sparsifySV+ . S.vr+ . fmap (bool 0 1 . (>= 0))+ . S.toDenseListSV+ . foldl1' S.concatSV+ . spectralNorm 2 1