[SPARK-9408] [PYSPARK] [MLLIB] Refactor linalg.py to /linalg

This is based on MechCoder 's PR https://github.com/apache/spark/pull/7731. Hopefully it could pass tests. MechCoder I tried to make minimal changes. If this passes Jenkins, we can merge this one first and then try to move `__init__.py` to `local.py` in a separate PR.

Closes #7731

Author: Xiangrui Meng <meng@databricks.com>

Closes #7746 from mengxr/SPARK-9408 and squashes the following commits:

0e05a3b [Xiangrui Meng] merge master
1135551 [Xiangrui Meng] add a comment for str(...)
c48cae0 [Xiangrui Meng] update tests
173a805 [Xiangrui Meng] move linalg.py to linalg/__init__.py

1 files changed, 1162 insertions, 0 deletions

diff --git a/python/pyspark/mllib/linalg/__init__.py b/python/pyspark/mllib/linalg/__init__.py
new file mode 100644
index 0000000000..334dc8e38b
--- /dev/null
+++ b/python/pyspark/mllib/linalg/__init__.py
@@ -0,0 +1,1162 @@
+#
+# Licensed to the Apache Software Foundation (ASF) under one or more
+# contributor license agreements.  See the NOTICE file distributed with
+# this work for additional information regarding copyright ownership.
+# The ASF licenses this file to You under the Apache License, Version 2.0
+# (the "License"); you may not use this file except in compliance with
+# the License.  You may obtain a copy of the License at
+#
+#    http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+#
+
+"""
+MLlib utilities for linear algebra. For dense vectors, MLlib
+uses the NumPy C{array} type, so you can simply pass NumPy arrays
+around. For sparse vectors, users can construct a L{SparseVector}
+object from MLlib or pass SciPy C{scipy.sparse} column vectors if
+SciPy is available in their environment.
+"""
+
+import sys
+import array
+
+if sys.version >= '3':
+    basestring = str
+    xrange = range
+    import copyreg as copy_reg
+    long = int
+else:
+    from itertools import izip as zip
+    import copy_reg
+
+import numpy as np
+
+from pyspark.sql.types import UserDefinedType, StructField, StructType, ArrayType, DoubleType, \
+    IntegerType, ByteType, BooleanType
+
+
+__all__ = ['Vector', 'DenseVector', 'SparseVector', 'Vectors',
+           'Matrix', 'DenseMatrix', 'SparseMatrix', 'Matrices']
+
+
+if sys.version_info[:2] == (2, 7):
+    # speed up pickling array in Python 2.7
+    def fast_pickle_array(ar):
+        return array.array, (ar.typecode, ar.tostring())
+    copy_reg.pickle(array.array, fast_pickle_array)
+
+
+# Check whether we have SciPy. MLlib works without it too, but if we have it, some methods,
+# such as _dot and _serialize_double_vector, start to support scipy.sparse matrices.
+
+try:
+    import scipy.sparse
+    _have_scipy = True
+except:
+    # No SciPy in environment, but that's okay
+    _have_scipy = False
+
+
+def _convert_to_vector(l):
+    if isinstance(l, Vector):
+        return l
+    elif type(l) in (array.array, np.array, np.ndarray, list, tuple, xrange):
+        return DenseVector(l)
+    elif _have_scipy and scipy.sparse.issparse(l):
+        assert l.shape[1] == 1, "Expected column vector"
+        csc = l.tocsc()
+        return SparseVector(l.shape[0], csc.indices, csc.data)
+    else:
+        raise TypeError("Cannot convert type %s into Vector" % type(l))
+
+
+def _vector_size(v):
+    """
+    Returns the size of the vector.
+
+    >>> _vector_size([1., 2., 3.])
+    3
+    >>> _vector_size((1., 2., 3.))
+    3
+    >>> _vector_size(array.array('d', [1., 2., 3.]))
+    3
+    >>> _vector_size(np.zeros(3))
+    3
+    >>> _vector_size(np.zeros((3, 1)))
+    3
+    >>> _vector_size(np.zeros((1, 3)))
+    Traceback (most recent call last):
+        ...
+    ValueError: Cannot treat an ndarray of shape (1, 3) as a vector
+    """
+    if isinstance(v, Vector):
+        return len(v)
+    elif type(v) in (array.array, list, tuple, xrange):
+        return len(v)
+    elif type(v) == np.ndarray:
+        if v.ndim == 1 or (v.ndim == 2 and v.shape[1] == 1):
+            return len(v)
+        else:
+            raise ValueError("Cannot treat an ndarray of shape %s as a vector" % str(v.shape))
+    elif _have_scipy and scipy.sparse.issparse(v):
+        assert v.shape[1] == 1, "Expected column vector"
+        return v.shape[0]
+    else:
+        raise TypeError("Cannot treat type %s as a vector" % type(v))
+
+
+def _format_float(f, digits=4):
+    s = str(round(f, digits))
+    if '.' in s:
+        s = s[:s.index('.') + 1 + digits]
+    return s
+
+
+def _format_float_list(l):
+    return [_format_float(x) for x in l]
+
+
+class VectorUDT(UserDefinedType):
+    """
+    SQL user-defined type (UDT) for Vector.
+    """
+
+    @classmethod
+    def sqlType(cls):
+        return StructType([
+            StructField("type", ByteType(), False),
+            StructField("size", IntegerType(), True),
+            StructField("indices", ArrayType(IntegerType(), False), True),
+            StructField("values", ArrayType(DoubleType(), False), True)])
+
+    @classmethod
+    def module(cls):
+        return "pyspark.mllib.linalg"
+
+    @classmethod
+    def scalaUDT(cls):
+        return "org.apache.spark.mllib.linalg.VectorUDT"
+
+    def serialize(self, obj):
+        if isinstance(obj, SparseVector):
+            indices = [int(i) for i in obj.indices]
+            values = [float(v) for v in obj.values]
+            return (0, obj.size, indices, values)
+        elif isinstance(obj, DenseVector):
+            values = [float(v) for v in obj]
+            return (1, None, None, values)
+        else:
+            raise TypeError("cannot serialize %r of type %r" % (obj, type(obj)))
+
+    def deserialize(self, datum):
+        assert len(datum) == 4, \
+            "VectorUDT.deserialize given row with length %d but requires 4" % len(datum)
+        tpe = datum[0]
+        if tpe == 0:
+            return SparseVector(datum[1], datum[2], datum[3])
+        elif tpe == 1:
+            return DenseVector(datum[3])
+        else:
+            raise ValueError("do not recognize type %r" % tpe)
+
+    def simpleString(self):
+        return "vector"
+
+
+class MatrixUDT(UserDefinedType):
+    """
+    SQL user-defined type (UDT) for Matrix.
+    """
+
+    @classmethod
+    def sqlType(cls):
+        return StructType([
+            StructField("type", ByteType(), False),
+            StructField("numRows", IntegerType(), False),
+            StructField("numCols", IntegerType(), False),
+            StructField("colPtrs", ArrayType(IntegerType(), False), True),
+            StructField("rowIndices", ArrayType(IntegerType(), False), True),
+            StructField("values", ArrayType(DoubleType(), False), True),
+            StructField("isTransposed", BooleanType(), False)])
+
+    @classmethod
+    def module(cls):
+        return "pyspark.mllib.linalg"
+
+    @classmethod
+    def scalaUDT(cls):
+        return "org.apache.spark.mllib.linalg.MatrixUDT"
+
+    def serialize(self, obj):
+        if isinstance(obj, SparseMatrix):
+            colPtrs = [int(i) for i in obj.colPtrs]
+            rowIndices = [int(i) for i in obj.rowIndices]
+            values = [float(v) for v in obj.values]
+            return (0, obj.numRows, obj.numCols, colPtrs,
+                    rowIndices, values, bool(obj.isTransposed))
+        elif isinstance(obj, DenseMatrix):
+            values = [float(v) for v in obj.values]
+            return (1, obj.numRows, obj.numCols, None, None, values,
+                    bool(obj.isTransposed))
+        else:
+            raise TypeError("cannot serialize type %r" % (type(obj)))
+
+    def deserialize(self, datum):
+        assert len(datum) == 7, \
+            "MatrixUDT.deserialize given row with length %d but requires 7" % len(datum)
+        tpe = datum[0]
+        if tpe == 0:
+            return SparseMatrix(*datum[1:])
+        elif tpe == 1:
+            return DenseMatrix(datum[1], datum[2], datum[5], datum[6])
+        else:
+            raise ValueError("do not recognize type %r" % tpe)
+
+    def simpleString(self):
+        return "matrix"
+
+
+class Vector(object):
+
+    __UDT__ = VectorUDT()
+
+    """
+    Abstract class for DenseVector and SparseVector
+    """
+    def toArray(self):
+        """
+        Convert the vector into an numpy.ndarray
+        :return: numpy.ndarray
+        """
+        raise NotImplementedError
+
+
+class DenseVector(Vector):
+    """
+    A dense vector represented by a value array. We use numpy array for
+    storage and arithmetics will be delegated to the underlying numpy
+    array.
+
+    >>> v = Vectors.dense([1.0, 2.0])
+    >>> u = Vectors.dense([3.0, 4.0])
+    >>> v + u
+    DenseVector([4.0, 6.0])
+    >>> 2 - v
+    DenseVector([1.0, 0.0])
+    >>> v / 2
+    DenseVector([0.5, 1.0])
+    >>> v * u
+    DenseVector([3.0, 8.0])
+    >>> u / v
+    DenseVector([3.0, 2.0])
+    >>> u % 2
+    DenseVector([1.0, 0.0])
+    """
+    def __init__(self, ar):
+        if isinstance(ar, bytes):
+            ar = np.frombuffer(ar, dtype=np.float64)
+        elif not isinstance(ar, np.ndarray):
+            ar = np.array(ar, dtype=np.float64)
+        if ar.dtype != np.float64:
+            ar = ar.astype(np.float64)
+        self.array = ar
+
+    @staticmethod
+    def parse(s):
+        """
+        Parse string representation back into the DenseVector.
+
+        >>> DenseVector.parse(' [ 0.0,1.0,2.0,  3.0]')
+        DenseVector([0.0, 1.0, 2.0, 3.0])
+        """
+        start = s.find('[')
+        if start == -1:
+            raise ValueError("Array should start with '['.")
+        end = s.find(']')
+        if end == -1:
+            raise ValueError("Array should end with ']'.")
+        s = s[start + 1: end]
+
+        try:
+            values = [float(val) for val in s.split(',')]
+        except ValueError:
+            raise ValueError("Unable to parse values from %s" % s)
+        return DenseVector(values)
+
+    def __reduce__(self):
+        return DenseVector, (self.array.tostring(),)
+
+    def numNonzeros(self):
+        return np.count_nonzero(self.array)
+
+    def norm(self, p):
+        """
+        Calculte the norm of a DenseVector.
+
+        >>> a = DenseVector([0, -1, 2, -3])
+        >>> a.norm(2)
+        3.7...
+        >>> a.norm(1)
+        6.0
+        """
+        return np.linalg.norm(self.array, p)
+
+    def dot(self, other):
+        """
+        Compute the dot product of two Vectors. We support
+        (Numpy array, list, SparseVector, or SciPy sparse)
+        and a target NumPy array that is either 1- or 2-dimensional.
+        Equivalent to calling numpy.dot of the two vectors.
+
+        >>> dense = DenseVector(array.array('d', [1., 2.]))
+        >>> dense.dot(dense)
+        5.0
+        >>> dense.dot(SparseVector(2, [0, 1], [2., 1.]))
+        4.0
+        >>> dense.dot(range(1, 3))
+        5.0
+        >>> dense.dot(np.array(range(1, 3)))
+        5.0
+        >>> dense.dot([1.,])
+        Traceback (most recent call last):
+            ...
+        AssertionError: dimension mismatch
+        >>> dense.dot(np.reshape([1., 2., 3., 4.], (2, 2), order='F'))
+        array([  5.,  11.])
+        >>> dense.dot(np.reshape([1., 2., 3.], (3, 1), order='F'))
+        Traceback (most recent call last):
+            ...
+        AssertionError: dimension mismatch
+        """
+        if type(other) == np.ndarray:
+            if other.ndim > 1:
+                assert len(self) == other.shape[0], "dimension mismatch"
+            return np.dot(self.array, other)
+        elif _have_scipy and scipy.sparse.issparse(other):
+            assert len(self) == other.shape[0], "dimension mismatch"
+            return other.transpose().dot(self.toArray())
+        else:
+            assert len(self) == _vector_size(other), "dimension mismatch"
+            if isinstance(other, SparseVector):
+                return other.dot(self)
+            elif isinstance(other, Vector):
+                return np.dot(self.toArray(), other.toArray())
+            else:
+                return np.dot(self.toArray(), other)
+
+    def squared_distance(self, other):
+        """
+        Squared distance of two Vectors.
+
+        >>> dense1 = DenseVector(array.array('d', [1., 2.]))
+        >>> dense1.squared_distance(dense1)
+        0.0
+        >>> dense2 = np.array([2., 1.])
+        >>> dense1.squared_distance(dense2)
+        2.0
+        >>> dense3 = [2., 1.]
+        >>> dense1.squared_distance(dense3)
+        2.0
+        >>> sparse1 = SparseVector(2, [0, 1], [2., 1.])
+        >>> dense1.squared_distance(sparse1)
+        2.0
+        >>> dense1.squared_distance([1.,])
+        Traceback (most recent call last):
+            ...
+        AssertionError: dimension mismatch
+        >>> dense1.squared_distance(SparseVector(1, [0,], [1.,]))
+        Traceback (most recent call last):
+            ...
+        AssertionError: dimension mismatch
+        """
+        assert len(self) == _vector_size(other), "dimension mismatch"
+        if isinstance(other, SparseVector):
+            return other.squared_distance(self)
+        elif _have_scipy and scipy.sparse.issparse(other):
+            return _convert_to_vector(other).squared_distance(self)
+
+        if isinstance(other, Vector):
+            other = other.toArray()
+        elif not isinstance(other, np.ndarray):
+            other = np.array(other)
+        diff = self.toArray() - other
+        return np.dot(diff, diff)
+
+    def toArray(self):
+        return self.array
+
+    def __getitem__(self, item):
+        return self.array[item]
+
+    def __len__(self):
+        return len(self.array)
+
+    def __str__(self):
+        return "[" + ",".join([str(v) for v in self.array]) + "]"
+
+    def __repr__(self):
+        return "DenseVector([%s])" % (', '.join(_format_float(i) for i in self.array))
+
+    def __eq__(self, other):
+        return isinstance(other, DenseVector) and np.array_equal(self.array, other.array)
+
+    def __ne__(self, other):
+        return not self == other
+
+    def __getattr__(self, item):
+        return getattr(self.array, item)
+
+    def _delegate(op):
+        def func(self, other):
+            if isinstance(other, DenseVector):
+                other = other.array
+            return DenseVector(getattr(self.array, op)(other))
+        return func
+
+    __neg__ = _delegate("__neg__")
+    __add__ = _delegate("__add__")
+    __sub__ = _delegate("__sub__")
+    __mul__ = _delegate("__mul__")
+    __div__ = _delegate("__div__")
+    __truediv__ = _delegate("__truediv__")
+    __mod__ = _delegate("__mod__")
+    __radd__ = _delegate("__radd__")
+    __rsub__ = _delegate("__rsub__")
+    __rmul__ = _delegate("__rmul__")
+    __rdiv__ = _delegate("__rdiv__")
+    __rtruediv__ = _delegate("__rtruediv__")
+    __rmod__ = _delegate("__rmod__")
+
+
+class SparseVector(Vector):
+    """
+    A simple sparse vector class for passing data to MLlib. Users may
+    alternatively pass SciPy's {scipy.sparse} data types.
+    """
+    def __init__(self, size, *args):
+        """
+        Create a sparse vector, using either a dictionary, a list of
+        (index, value) pairs, or two separate arrays of indices and
+        values (sorted by index).
+
+        :param size: Size of the vector.
+        :param args: Active entries, as a dictionary {index: value, ...},
+          a list of tuples [(index, value), ...], or a list of strictly i
+          ncreasing indices and a list of corresponding values [index, ...],
+          [value, ...]. Inactive entries are treated as zeros.
+
+        >>> SparseVector(4, {1: 1.0, 3: 5.5})
+        SparseVector(4, {1: 1.0, 3: 5.5})
+        >>> SparseVector(4, [(1, 1.0), (3, 5.5)])
+        SparseVector(4, {1: 1.0, 3: 5.5})
+        >>> SparseVector(4, [1, 3], [1.0, 5.5])
+        SparseVector(4, {1: 1.0, 3: 5.5})
+        """
+        self.size = int(size)
+        """ Size of the vector. """
+        assert 1 <= len(args) <= 2, "must pass either 2 or 3 arguments"
+        if len(args) == 1:
+            pairs = args[0]
+            if type(pairs) == dict:
+                pairs = pairs.items()
+            pairs = sorted(pairs)
+            self.indices = np.array([p[0] for p in pairs], dtype=np.int32)
+            """ A list of indices corresponding to active entries. """
+            self.values = np.array([p[1] for p in pairs], dtype=np.float64)
+            """ A list of values corresponding to active entries. """
+        else:
+            if isinstance(args[0], bytes):
+                assert isinstance(args[1], bytes), "values should be string too"
+                if args[0]:
+                    self.indices = np.frombuffer(args[0], np.int32)
+                    self.values = np.frombuffer(args[1], np.float64)
+                else:
+                    # np.frombuffer() doesn't work well with empty string in older version
+                    self.indices = np.array([], dtype=np.int32)
+                    self.values = np.array([], dtype=np.float64)
+            else:
+                self.indices = np.array(args[0], dtype=np.int32)
+                self.values = np.array(args[1], dtype=np.float64)
+            assert len(self.indices) == len(self.values), "index and value arrays not same length"
+            for i in xrange(len(self.indices) - 1):
+                if self.indices[i] >= self.indices[i + 1]:
+                    raise TypeError("indices array must be sorted")
+
+    def numNonzeros(self):
+        return np.count_nonzero(self.values)
+
+    def norm(self, p):
+        """
+        Calculte the norm of a SparseVector.
+
+        >>> a = SparseVector(4, [0, 1], [3., -4.])
+        >>> a.norm(1)
+        7.0
+        >>> a.norm(2)
+        5.0
+        """
+        return np.linalg.norm(self.values, p)
+
+    def __reduce__(self):
+        return (
+            SparseVector,
+            (self.size, self.indices.tostring(), self.values.tostring()))
+
+    @staticmethod
+    def parse(s):
+        """
+        Parse string representation back into the DenseVector.
+
+        >>> SparseVector.parse(' (4, [0,1 ],[ 4.0,5.0] )')
+        SparseVector(4, {0: 4.0, 1: 5.0})
+        """
+        start = s.find('(')
+        if start == -1:
+            raise ValueError("Tuple should start with '('")
+        end = s.find(')')
+        if start == -1:
+            raise ValueError("Tuple should end with ')'")
+        s = s[start + 1: end].strip()
+
+        size = s[: s.find(',')]
+        try:
+            size = int(size)
+        except ValueError:
+            raise ValueError("Cannot parse size %s." % size)
+
+        ind_start = s.find('[')
+        if ind_start == -1:
+            raise ValueError("Indices array should start with '['.")
+        ind_end = s.find(']')
+        if ind_end == -1:
+            raise ValueError("Indices array should end with ']'")
+        new_s = s[ind_start + 1: ind_end]
+        ind_list = new_s.split(',')
+        try:
+            indices = [int(ind) for ind in ind_list]
+        except ValueError:
+            raise ValueError("Unable to parse indices from %s." % new_s)
+        s = s[ind_end + 1:].strip()
+
+        val_start = s.find('[')
+        if val_start == -1:
+            raise ValueError("Values array should start with '['.")
+        val_end = s.find(']')
+        if val_end == -1:
+            raise ValueError("Values array should end with ']'.")
+        val_list = s[val_start + 1: val_end].split(',')
+        try:
+            values = [float(val) for val in val_list]
+        except ValueError:
+            raise ValueError("Unable to parse values from %s." % s)
+        return SparseVector(size, indices, values)
+
+    def dot(self, other):
+        """
+        Dot product with a SparseVector or 1- or 2-dimensional Numpy array.
+
+        >>> a = SparseVector(4, [1, 3], [3.0, 4.0])
+        >>> a.dot(a)
+        25.0
+        >>> a.dot(array.array('d', [1., 2., 3., 4.]))
+        22.0
+        >>> b = SparseVector(4, [2], [1.0])
+        >>> a.dot(b)
+        0.0
+        >>> a.dot(np.array([[1, 1], [2, 2], [3, 3], [4, 4]]))
+        array([ 22.,  22.])
+        >>> a.dot([1., 2., 3.])
+        Traceback (most recent call last):
+            ...
+        AssertionError: dimension mismatch
+        >>> a.dot(np.array([1., 2.]))
+        Traceback (most recent call last):
+            ...
+        AssertionError: dimension mismatch
+        >>> a.dot(DenseVector([1., 2.]))
+        Traceback (most recent call last):
+            ...
+        AssertionError: dimension mismatch
+        >>> a.dot(np.zeros((3, 2)))
+        Traceback (most recent call last):
+            ...
+        AssertionError: dimension mismatch
+        """
+
+        if isinstance(other, np.ndarray):
+            if other.ndim not in [2, 1]:
+                raise ValueError("Cannot call dot with %d-dimensional array" % other.ndim)
+            assert len(self) == other.shape[0], "dimension mismatch"
+            return np.dot(self.values, other[self.indices])
+
+        assert len(self) == _vector_size(other), "dimension mismatch"
+
+        if isinstance(other, DenseVector):
+            return np.dot(other.array[self.indices], self.values)
+
+        elif isinstance(other, SparseVector):
+            # Find out common indices.
+            self_cmind = np.in1d(self.indices, other.indices, assume_unique=True)
+            self_values = self.values[self_cmind]
+            if self_values.size == 0:
+                return 0.0
+            else:
+                other_cmind = np.in1d(other.indices, self.indices, assume_unique=True)
+                return np.dot(self_values, other.values[other_cmind])
+
+        else:
+            return self.dot(_convert_to_vector(other))
+
+    def squared_distance(self, other):
+        """
+        Squared distance from a SparseVector or 1-dimensional NumPy array.
+
+        >>> a = SparseVector(4, [1, 3], [3.0, 4.0])
+        >>> a.squared_distance(a)
+        0.0
+        >>> a.squared_distance(array.array('d', [1., 2., 3., 4.]))
+        11.0
+        >>> a.squared_distance(np.array([1., 2., 3., 4.]))
+        11.0
+        >>> b = SparseVector(4, [2], [1.0])
+        >>> a.squared_distance(b)
+        26.0
+        >>> b.squared_distance(a)
+        26.0
+        >>> b.squared_distance([1., 2.])
+        Traceback (most recent call last):
+            ...
+        AssertionError: dimension mismatch
+        >>> b.squared_distance(SparseVector(3, [1,], [1.0,]))
+        Traceback (most recent call last):
+            ...
+        AssertionError: dimension mismatch
+        """
+        assert len(self) == _vector_size(other), "dimension mismatch"
+
+        if isinstance(other, np.ndarray) or isinstance(other, DenseVector):
+            if isinstance(other, np.ndarray) and other.ndim != 1:
+                raise Exception("Cannot call squared_distance with %d-dimensional array" %
+                                other.ndim)
+            if isinstance(other, DenseVector):
+                other = other.array
+            sparse_ind = np.zeros(other.size, dtype=bool)
+            sparse_ind[self.indices] = True
+            dist = other[sparse_ind] - self.values
+            result = np.dot(dist, dist)
+
+            other_ind = other[~sparse_ind]
+            result += np.dot(other_ind, other_ind)
+            return result
+
+        elif isinstance(other, SparseVector):
+            result = 0.0
+            i, j = 0, 0
+            while i < len(self.indices) and j < len(other.indices):
+                if self.indices[i] == other.indices[j]:
+                    diff = self.values[i] - other.values[j]
+                    result += diff * diff
+                    i += 1
+                    j += 1
+                elif self.indices[i] < other.indices[j]:
+                    result += self.values[i] * self.values[i]
+                    i += 1
+                else:
+                    result += other.values[j] * other.values[j]
+                    j += 1
+            while i < len(self.indices):
+                result += self.values[i] * self.values[i]
+                i += 1
+            while j < len(other.indices):
+                result += other.values[j] * other.values[j]
+                j += 1
+            return result
+        else:
+            return self.squared_distance(_convert_to_vector(other))
+
+    def toArray(self):
+        """
+        Returns a copy of this SparseVector as a 1-dimensional NumPy array.
+        """
+        arr = np.zeros((self.size,), dtype=np.float64)
+        arr[self.indices] = self.values
+        return arr
+
+    def __len__(self):
+        return self.size
+
+    def __str__(self):
+        inds = "[" + ",".join([str(i) for i in self.indices]) + "]"
+        vals = "[" + ",".join([str(v) for v in self.values]) + "]"
+        return "(" + ",".join((str(self.size), inds, vals)) + ")"
+
+    def __repr__(self):
+        inds = self.indices
+        vals = self.values
+        entries = ", ".join(["{0}: {1}".format(inds[i], _format_float(vals[i]))
+                             for i in xrange(len(inds))])
+        return "SparseVector({0}, {{{1}}})".format(self.size, entries)
+
+    def __eq__(self, other):
+        """
+        Test SparseVectors for equality.
+
+        >>> v1 = SparseVector(4, [(1, 1.0), (3, 5.5)])
+        >>> v2 = SparseVector(4, [(1, 1.0), (3, 5.5)])
+        >>> v1 == v2
+        True
+        >>> v1 != v2
+        False
+        """
+        return (isinstance(other, self.__class__)
+                and other.size == self.size
+                and np.array_equal(other.indices, self.indices)
+                and np.array_equal(other.values, self.values))
+
+    def __getitem__(self, index):
+        inds = self.indices
+        vals = self.values
+        if not isinstance(index, int):
+            raise TypeError(
+                "Indices must be of type integer, got type %s" % type(index))
+        if index < 0:
+            index += self.size
+        if index >= self.size or index < 0:
+            raise ValueError("Index %d out of bounds." % index)
+
+        insert_index = np.searchsorted(inds, index)
+        row_ind = inds[insert_index]
+        if row_ind == index:
+            return vals[insert_index]
+        return 0.
+
+    def __ne__(self, other):
+        return not self.__eq__(other)
+
+
+class Vectors(object):
+
+    """
+    Factory methods for working with vectors. Note that dense vectors
+    are simply represented as NumPy array objects, so there is no need
+    to covert them for use in MLlib. For sparse vectors, the factory
+    methods in this class create an MLlib-compatible type, or users
+    can pass in SciPy's C{scipy.sparse} column vectors.
+    """
+
+    @staticmethod
+    def sparse(size, *args):
+        """
+        Create a sparse vector, using either a dictionary, a list of
+        (index, value) pairs, or two separate arrays of indices and
+        values (sorted by index).
+
+        :param size: Size of the vector.
+        :param args: Non-zero entries, as a dictionary, list of tupes,
+                     or two sorted lists containing indices and values.
+
+        >>> Vectors.sparse(4, {1: 1.0, 3: 5.5})
+        SparseVector(4, {1: 1.0, 3: 5.5})
+        >>> Vectors.sparse(4, [(1, 1.0), (3, 5.5)])
+        SparseVector(4, {1: 1.0, 3: 5.5})
+        >>> Vectors.sparse(4, [1, 3], [1.0, 5.5])
+        SparseVector(4, {1: 1.0, 3: 5.5})
+        """
+        return SparseVector(size, *args)
+
+    @staticmethod
+    def dense(*elements):
+        """
+        Create a dense vector of 64-bit floats from a Python list or numbers.
+
+        >>> Vectors.dense([1, 2, 3])
+        DenseVector([1.0, 2.0, 3.0])
+        >>> Vectors.dense(1.0, 2.0)
+        DenseVector([1.0, 2.0])
+        """
+        if len(elements) == 1 and not isinstance(elements[0], (float, int, long)):
+            # it's list, numpy.array or other iterable object.
+            elements = elements[0]
+        return DenseVector(elements)
+
+    @staticmethod
+    def stringify(vector):
+        """
+        Converts a vector into a string, which can be recognized by
+        Vectors.parse().
+
+        >>> Vectors.stringify(Vectors.sparse(2, [1], [1.0]))
+        '(2,[1],[1.0])'
+        >>> Vectors.stringify(Vectors.dense([0.0, 1.0]))
+        '[0.0,1.0]'
+        """
+        return str(vector)
+
+    @staticmethod
+    def squared_distance(v1, v2):
+        """
+        Squared distance between two vectors.
+        a and b can be of type SparseVector, DenseVector, np.ndarray
+        or array.array.
+
+        >>> a = Vectors.sparse(4, [(0, 1), (3, 4)])
+        >>> b = Vectors.dense([2, 5, 4, 1])
+        >>> a.squared_distance(b)
+        51.0
+        """
+        v1, v2 = _convert_to_vector(v1), _convert_to_vector(v2)
+        return v1.squared_distance(v2)
+
+    @staticmethod
+    def norm(vector, p):
+        """
+        Find norm of the given vector.
+        """
+        return _convert_to_vector(vector).norm(p)
+
+    @staticmethod
+    def parse(s):
+        """Parse a string representation back into the Vector.
+
+        >>> Vectors.parse('[2,1,2 ]')
+        DenseVector([2.0, 1.0, 2.0])
+        >>> Vectors.parse(' ( 100,  [0],  [2])')
+        SparseVector(100, {0: 2.0})
+        """
+        if s.find('(') == -1 and s.find('[') != -1:
+            return DenseVector.parse(s)
+        elif s.find('(') != -1:
+            return SparseVector.parse(s)
+        else:
+            raise ValueError(
+                "Cannot find tokens '[' or '(' from the input string.")
+
+    @staticmethod
+    def zeros(size):
+        return DenseVector(np.zeros(size))
+
+
+class Matrix(object):
+
+    __UDT__ = MatrixUDT()
+
+    """
+    Represents a local matrix.
+    """
+    def __init__(self, numRows, numCols, isTransposed=False):
+        self.numRows = numRows
+        self.numCols = numCols
+        self.isTransposed = isTransposed
+
+    def toArray(self):
+        """
+        Returns its elements in a NumPy ndarray.
+        """
+        raise NotImplementedError
+
+    @staticmethod
+    def _convert_to_array(array_like, dtype):
+        """
+        Convert Matrix attributes which are array-like or buffer to array.
+        """
+        if isinstance(array_like, bytes):
+            return np.frombuffer(array_like, dtype=dtype)
+        return np.asarray(array_like, dtype=dtype)
+
+
+class DenseMatrix(Matrix):
+    """
+    Column-major dense matrix.
+    """
+    def __init__(self, numRows, numCols, values, isTransposed=False):
+        Matrix.__init__(self, numRows, numCols, isTransposed)
+        values = self._convert_to_array(values, np.float64)
+        assert len(values) == numRows * numCols
+        self.values = values
+
+    def __reduce__(self):
+        return DenseMatrix, (
+            self.numRows, self.numCols, self.values.tostring(),
+            int(self.isTransposed))
+
+    def __str__(self):
+        """
+        Pretty printing of a DenseMatrix
+
+        >>> dm = DenseMatrix(2, 2, range(4))
+        >>> print(dm)
+        DenseMatrix([[ 0.,  2.],
+                     [ 1.,  3.]])
+        >>> dm = DenseMatrix(2, 2, range(4), isTransposed=True)
+        >>> print(dm)
+        DenseMatrix([[ 0.,  1.],
+                     [ 2.,  3.]])
+        """
+        # Inspired by __repr__ in scipy matrices.
+        array_lines = repr(self.toArray()).splitlines()
+
+        # We need to adjust six spaces which is the difference in number
+        # of letters between "DenseMatrix" and "array"
+        x = '\n'.join([(" " * 6 + line) for line in array_lines[1:]])
+        return array_lines[0].replace("array", "DenseMatrix") + "\n" + x
+
+    def __repr__(self):
+        """
+        Representation of a DenseMatrix
+
+        >>> dm = DenseMatrix(2, 2, range(4))
+        >>> dm
+        DenseMatrix(2, 2, [0.0, 1.0, 2.0, 3.0], False)
+        """
+        # If the number of values are less than seventeen then return as it is.
+        # Else return first eight values and last eight values.
+        if len(self.values) < 17:
+            entries = _format_float_list(self.values)
+        else:
+            entries = (
+                _format_float_list(self.values[:8]) +
+                ["..."] +
+                _format_float_list(self.values[-8:])
+            )
+
+        entries = ", ".join(entries)
+        return "DenseMatrix({0}, {1}, [{2}], {3})".format(
+            self.numRows, self.numCols, entries, self.isTransposed)
+
+    def toArray(self):
+        """
+        Return an numpy.ndarray
+
+        >>> m = DenseMatrix(2, 2, range(4))
+        >>> m.toArray()
+        array([[ 0.,  2.],
+               [ 1.,  3.]])
+        """
+        if self.isTransposed:
+            return np.asfortranarray(
+                self.values.reshape((self.numRows, self.numCols)))
+        else:
+            return self.values.reshape((self.numRows, self.numCols), order='F')
+
+    def toSparse(self):
+        """Convert to SparseMatrix"""
+        if self.isTransposed:
+            values = np.ravel(self.toArray(), order='F')
+        else:
+            values = self.values
+        indices = np.nonzero(values)[0]
+        colCounts = np.bincount(indices // self.numRows)
+        colPtrs = np.cumsum(np.hstack(
+            (0, colCounts, np.zeros(self.numCols - colCounts.size))))
+        values = values[indices]
+        rowIndices = indices % self.numRows
+
+        return SparseMatrix(self.numRows, self.numCols, colPtrs, rowIndices, values)
+
+    def __getitem__(self, indices):
+        i, j = indices
+        if i < 0 or i >= self.numRows:
+            raise ValueError("Row index %d is out of range [0, %d)"
+                             % (i, self.numRows))
+        if j >= self.numCols or j < 0:
+            raise ValueError("Column index %d is out of range [0, %d)"
+                             % (j, self.numCols))
+
+        if self.isTransposed:
+            return self.values[i * self.numCols + j]
+        else:
+            return self.values[i + j * self.numRows]
+
+    def __eq__(self, other):
+        if (not isinstance(other, DenseMatrix) or
+                self.numRows != other.numRows or
+                self.numCols != other.numCols):
+            return False
+
+        self_values = np.ravel(self.toArray(), order='F')
+        other_values = np.ravel(other.toArray(), order='F')
+        return all(self_values == other_values)
+
+
+class SparseMatrix(Matrix):
+    """Sparse Matrix stored in CSC format."""
+    def __init__(self, numRows, numCols, colPtrs, rowIndices, values,
+                 isTransposed=False):
+        Matrix.__init__(self, numRows, numCols, isTransposed)
+        self.colPtrs = self._convert_to_array(colPtrs, np.int32)
+        self.rowIndices = self._convert_to_array(rowIndices, np.int32)
+        self.values = self._convert_to_array(values, np.float64)
+
+        if self.isTransposed:
+            if self.colPtrs.size != numRows + 1:
+                raise ValueError("Expected colPtrs of size %d, got %d."
+                                 % (numRows + 1, self.colPtrs.size))
+        else:
+            if self.colPtrs.size != numCols + 1:
+                raise ValueError("Expected colPtrs of size %d, got %d."
+                                 % (numCols + 1, self.colPtrs.size))
+        if self.rowIndices.size != self.values.size:
+            raise ValueError("Expected rowIndices of length %d, got %d."
+                             % (self.rowIndices.size, self.values.size))
+
+    def __str__(self):
+        """
+        Pretty printing of a SparseMatrix
+
+        >>> sm1 = SparseMatrix(2, 2, [0, 2, 3], [0, 1, 1], [2, 3, 4])
+        >>> print(sm1)
+        2 X 2 CSCMatrix
+        (0,0) 2.0
+        (1,0) 3.0
+        (1,1) 4.0
+        >>> sm1 = SparseMatrix(2, 2, [0, 2, 3], [0, 1, 1], [2, 3, 4], True)
+        >>> print(sm1)
+        2 X 2 CSRMatrix
+        (0,0) 2.0
+        (0,1) 3.0
+        (1,1) 4.0
+        """
+        spstr = "{0} X {1} ".format(self.numRows, self.numCols)
+        if self.isTransposed:
+            spstr += "CSRMatrix\n"
+        else:
+            spstr += "CSCMatrix\n"
+
+        cur_col = 0
+        smlist = []
+
+        # Display first 16 values.
+        if len(self.values) <= 16:
+            zipindval = zip(self.rowIndices, self.values)
+        else:
+            zipindval = zip(self.rowIndices[:16], self.values[:16])
+        for i, (rowInd, value) in enumerate(zipindval):
+            if self.colPtrs[cur_col + 1] <= i:
+                cur_col += 1
+            if self.isTransposed:
+                smlist.append('({0},{1}) {2}'.format(
+                    cur_col, rowInd, _format_float(value)))
+            else:
+                smlist.append('({0},{1}) {2}'.format(
+                    rowInd, cur_col, _format_float(value)))
+        spstr += "\n".join(smlist)
+
+        if len(self.values) > 16:
+            spstr += "\n.." * 2
+        return spstr
+
+    def __repr__(self):
+        """
+        Representation of a SparseMatrix
+
+        >>> sm1 = SparseMatrix(2, 2, [0, 2, 3], [0, 1, 1], [2, 3, 4])
+        >>> sm1
+        SparseMatrix(2, 2, [0, 2, 3], [0, 1, 1], [2.0, 3.0, 4.0], False)
+        """
+        rowIndices = list(self.rowIndices)
+        colPtrs = list(self.colPtrs)
+
+        if len(self.values) <= 16:
+            values = _format_float_list(self.values)
+
+        else:
+            values = (
+                _format_float_list(self.values[:8]) +
+                ["..."] +
+                _format_float_list(self.values[-8:])
+            )
+            rowIndices = rowIndices[:8] + ["..."] + rowIndices[-8:]
+
+        if len(self.colPtrs) > 16:
+            colPtrs = colPtrs[:8] + ["..."] + colPtrs[-8:]
+
+        values = ", ".join(values)
+        rowIndices = ", ".join([str(ind) for ind in rowIndices])
+        colPtrs = ", ".join([str(ptr) for ptr in colPtrs])
+        return "SparseMatrix({0}, {1}, [{2}], [{3}], [{4}], {5})".format(
+            self.numRows, self.numCols, colPtrs, rowIndices,
+            values, self.isTransposed)
+
+    def __reduce__(self):
+        return SparseMatrix, (
+            self.numRows, self.numCols, self.colPtrs.tostring(),
+            self.rowIndices.tostring(), self.values.tostring(),
+            int(self.isTransposed))
+
+    def __getitem__(self, indices):
+        i, j = indices
+        if i < 0 or i >= self.numRows:
+            raise ValueError("Row index %d is out of range [0, %d)"
+                             % (i, self.numRows))
+        if j < 0 or j >= self.numCols:
+            raise ValueError("Column index %d is out of range [0, %d)"
+                             % (j, self.numCols))
+
+        # If a CSR matrix is given, then the row index should be searched
+        # for in ColPtrs, and the column index should be searched for in the
+        # corresponding slice obtained from rowIndices.
+        if self.isTransposed:
+            j, i = i, j
+
+        colStart = self.colPtrs[j]
+        colEnd = self.colPtrs[j + 1]
+        nz = self.rowIndices[colStart: colEnd]
+        ind = np.searchsorted(nz, i) + colStart
+        if ind < colEnd and self.rowIndices[ind] == i:
+            return self.values[ind]
+        else:
+            return 0.0
+
+    def toArray(self):
+        """
+        Return an numpy.ndarray
+        """
+        A = np.zeros((self.numRows, self.numCols), dtype=np.float64, order='F')
+        for k in xrange(self.colPtrs.size - 1):
+            startptr = self.colPtrs[k]
+            endptr = self.colPtrs[k + 1]
+            if self.isTransposed:
+                A[k, self.rowIndices[startptr:endptr]] = self.values[startptr:endptr]
+            else:
+                A[self.rowIndices[startptr:endptr], k] = self.values[startptr:endptr]
+        return A
+
+    def toDense(self):
+        densevals = np.ravel(self.toArray(), order='F')
+        return DenseMatrix(self.numRows, self.numCols, densevals)
+
+    # TODO: More efficient implementation:
+    def __eq__(self, other):
+        return np.all(self.toArray() == other.toArray())
+
+
+class Matrices(object):
+    @staticmethod
+    def dense(numRows, numCols, values):
+        """
+        Create a DenseMatrix
+        """
+        return DenseMatrix(numRows, numCols, values)
+
+    @staticmethod
+    def sparse(numRows, numCols, colPtrs, rowIndices, values):
+        """
+        Create a SparseMatrix
+        """
+        return SparseMatrix(numRows, numCols, colPtrs, rowIndices, values)
+
+
+def _test():
+    import doctest
+    (failure_count, test_count) = doctest.testmod(optionflags=doctest.ELLIPSIS)
+    if failure_count:
+        exit(-1)
+
+if __name__ == "__main__":
+    _test()