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Similarity API

Basic Similarities

Raw Dot Product

Compute dot product similarity between rows of matrix1 and columns of matrix2.

Parameters:

Name Type Description Default
matrix1 spmatrix

Input sparse matrix (e.g., user-item or item-user).

 required
matrix2 Optional[spmatrix]

Optional second matrix. If None, uses matrix1.T.

 None
k int

Number of top-k items per row.

 100
shrink float

Shrinkage value that prevents instability when normalizations are small.

 0.0
shrink_type Literal['stabilized', 'bayesian', 'additive']

Type of shrinkage: 'stabilized', 'bayesian', 'additive'.

 'stabilized'
threshold float

Minimum similarity value to retain.

 0.0
binary bool

Whether to binarize the input matrix before similarity computation.

 False
target_rows Optional[Union[list[int], ndarray]]

List or array of row indices to compute. If None, computes all.

 None
target_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to include before top-k. Can be a list or sparse mask matrix.

 None
filter_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to exclude before top-k. Can be a list or sparse mask matrix.

 None
verbose bool

Whether to show a progress bar.

 True
format_output Literal['csr', 'coo']

Output format: 'csr' or 'coo'. Use 'coo' on Windows.

 'coo'
num_threads int

Number of threads to use (0 means all available cores).

 0

Returns:

Type Description
spmatrix

A sparse matrix of shape (n_rows, n_cols) in the specified format,
spmatrix

containing the top-k dot product similarities.
Source code in similaripy/similarity.py 
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def dot_product(
    matrix1: spmatrix,
    matrix2: Optional[spmatrix] = None,
    k: int = 100,
    shrink: float = 0.0,
    shrink_type: Literal['stabilized', 'bayesian', 'additive'] = 'stabilized',
    threshold: float = 0.0,
    binary: bool = False,
    target_rows: Optional[Union[list[int], np.ndarray]] = None,
    target_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    filter_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    verbose: bool = True,
    format_output: Literal['csr', 'coo'] = 'coo',
    num_threads: int = 0
) -> spmatrix:
    """
    Compute dot product similarity between rows of matrix1 and columns of matrix2.

    Args:
        matrix1: Input sparse matrix (e.g., user-item or item-user).
        matrix2: Optional second matrix. If None, uses matrix1.T.
        k: Number of top-k items per row.
        shrink: Shrinkage value that prevents instability when normalizations are small.
        shrink_type: Type of shrinkage: 'stabilized', 'bayesian', 'additive'.
        threshold: Minimum similarity value to retain.
        binary: Whether to binarize the input matrix before similarity computation.
        target_rows: List or array of row indices to compute. If None, computes all.
        target_cols: Columns to include before top-k. Can be a list or sparse mask matrix.
        filter_cols: Columns to exclude before top-k. Can be a list or sparse mask matrix.
        verbose: Whether to show a progress bar.
        format_output: Output format: 'csr' or 'coo'. Use 'coo' on Windows.
        num_threads: Number of threads to use (0 means all available cores).

    Returns:
        A sparse matrix of shape (n_rows, n_cols) in the specified format,
        containing the top-k dot product similarities.
    """
    stabilized_shrink, bayesian_shrink, additive_shrink = __get_shrink_values__(shrink, shrink_type)
    return _sim.s_plus(
        matrix1, matrix2=matrix2,
        k=k,
        stabilized_shrink=stabilized_shrink,
        bayesian_shrink=bayesian_shrink,
        additive_shrink=additive_shrink,
        threshold=threshold,
        binary=binary,
        target_rows=target_rows,
        target_cols=target_cols,
        filter_cols=filter_cols,
        verbose=verbose,
        format_output=format_output,
        num_threads=num_threads
    )
Cosine Similarity

Compute cosine similarity between sparse vectors.

Parameters:

Name Type Description Default
matrix1 spmatrix

Input sparse matrix.

 required
matrix2 Optional[spmatrix]

Optional second matrix. If None, uses matrix1.T.

 None
k int

Number of top-k items per row to keep.

 100
shrink float

Shrinkage value that prevents instability when normalizations are small.

 0.0
shrink_type Literal['stabilized', 'bayesian', 'additive']

Type of shrinkage: 'stabilized', 'bayesian', 'additive'.

 'stabilized'
threshold float

Minimum similarity value to retain.

 0.0
binary bool

Whether to binarize the input matrix before computation.

 False
target_rows Optional[Union[list[int], ndarray]]

List or array of row indices to compute. If None, computes all.

 None
target_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to include before top-k. Can be a list or sparse mask matrix.

 None
filter_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to exclude before top-k. Can be a list or sparse mask matrix.

 None
verbose bool

Whether to show a progress bar.

 True
format_output Literal['csr', 'coo']

Output format, either 'csr' or 'coo'. Use 'coo' on Windows.

 'coo'
num_threads int

Number of threads to use (0 = all available cores).

 0

Returns:

Type Description
spmatrix

A sparse matrix of top-k cosine similarities in the specified format.
Source code in similaripy/similarity.py 
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def cosine(
    matrix1: spmatrix,
    matrix2: Optional[spmatrix] = None,
    k: int = 100,
    shrink: float = 0.0,
    shrink_type: Literal['stabilized', 'bayesian', 'additive'] = 'stabilized',
    threshold: float = 0.0,
    binary: bool = False,
    target_rows: Optional[Union[list[int], np.ndarray]] = None,
    target_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    filter_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    verbose: bool = True,
    format_output: Literal['csr', 'coo'] = 'coo',
    num_threads: int = 0
) -> spmatrix:
    """
    Compute cosine similarity between sparse vectors.

    Args:
        matrix1: Input sparse matrix.
        matrix2: Optional second matrix. If None, uses matrix1.T.
        k: Number of top-k items per row to keep.
        shrink: Shrinkage value that prevents instability when normalizations are small.
        shrink_type: Type of shrinkage: 'stabilized', 'bayesian', 'additive'.
        threshold: Minimum similarity value to retain.
        binary: Whether to binarize the input matrix before computation.
        target_rows: List or array of row indices to compute. If None, computes all.
        target_cols: Columns to include before top-k. Can be a list or sparse mask matrix.
        filter_cols: Columns to exclude before top-k. Can be a list or sparse mask matrix.
        verbose: Whether to show a progress bar.
        format_output: Output format, either 'csr' or 'coo'. Use 'coo' on Windows.
        num_threads: Number of threads to use (0 = all available cores).

    Returns:
        A sparse matrix of top-k cosine similarities in the specified format.
    """
    stabilized_shrink, bayesian_shrink, additive_shrink = __get_shrink_values__(shrink, shrink_type)
    return _sim.s_plus(
        matrix1, matrix2=matrix2,
        l2=1,
        c1=0.5, c2=0.5,
        k=k,
        stabilized_shrink=stabilized_shrink,
        bayesian_shrink=bayesian_shrink,
        additive_shrink=additive_shrink,
        threshold=threshold,
        binary=binary,
        target_rows=target_rows,
        target_cols=target_cols,
        filter_cols=filter_cols,
        verbose=verbose,
        format_output=format_output,
        num_threads=num_threads
    )
Asymmetric Cosine

Compute asymmetric cosine similarity.

Parameters:

Name Type Description Default
matrix1 spmatrix

Input sparse matrix (e.g., user-item or item-user).

 required
matrix2 Optional[spmatrix]

Optional second matrix. If None, uses matrix1.T.

 None
alpha float

Controls asymmetry in cosine weighting. alpha=1 weighs only matrix1; alpha=0.5 is symmetric.

 0.5
k int

Number of top-k items per row to keep.

 100
shrink float

Shrinkage value that prevents instability when normalizations are small.

 0.0
shrink_type Literal['stabilized', 'bayesian', 'additive']

Type of shrinkage: 'stabilized', 'bayesian', 'additive'.

 'stabilized'
threshold float

Minimum similarity value to retain.

 0.0
binary bool

Whether to binarize the input matrix before computation.

 False
target_rows Optional[Union[list[int], ndarray]]

List or array of row indices to compute. If None, computes all.

 None
target_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to include before top-k. Can be a list or sparse mask matrix.

 None
filter_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to exclude before top-k. Can be a list or sparse mask matrix.

 None
verbose bool

Whether to show a progress bar.

 True
format_output Literal['csr', 'coo']

Output format, either 'csr' or 'coo'. Use 'coo' on Windows.

 'coo'
num_threads int

Number of threads to use (0 = all available cores).

 0

Returns:

Type Description
spmatrix

A sparse matrix of shape (n_rows, n_cols) containing the top-k
spmatrix

asymmetric cosine similarities in the specified format.
Source code in similaripy/similarity.py 
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def asymmetric_cosine(
    matrix1: spmatrix,
    matrix2: Optional[spmatrix] = None,
    alpha: float = 0.5,
    k: int = 100,
    shrink: float = 0.0,
    shrink_type: Literal['stabilized', 'bayesian', 'additive'] = 'stabilized',
    threshold: float = 0.0,
    binary: bool = False,
    target_rows: Optional[Union[list[int], np.ndarray]] = None,
    target_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    filter_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    verbose: bool = True,
    format_output: Literal['csr', 'coo'] = 'coo',
    num_threads: int = 0
) -> spmatrix:
    """
    Compute asymmetric cosine similarity.

    Args:
        matrix1: Input sparse matrix (e.g., user-item or item-user).
        matrix2: Optional second matrix. If None, uses matrix1.T.
        alpha: Controls asymmetry in cosine weighting.
               `alpha=1` weighs only matrix1; `alpha=0.5` is symmetric.
        k: Number of top-k items per row to keep.
        shrink: Shrinkage value that prevents instability when normalizations are small.
        shrink_type: Type of shrinkage: 'stabilized', 'bayesian', 'additive'.
        threshold: Minimum similarity value to retain.
        binary: Whether to binarize the input matrix before computation.
        target_rows: List or array of row indices to compute. If None, computes all.
        target_cols: Columns to include before top-k. Can be a list or sparse mask matrix.
        filter_cols: Columns to exclude before top-k. Can be a list or sparse mask matrix.
        verbose: Whether to show a progress bar.
        format_output: Output format, either 'csr' or 'coo'. Use 'coo' on Windows.
        num_threads: Number of threads to use (0 = all available cores).

    Returns:
        A sparse matrix of shape (n_rows, n_cols) containing the top-k
        asymmetric cosine similarities in the specified format.
    """
    stabilized_shrink, bayesian_shrink, additive_shrink = __get_shrink_values__(shrink, shrink_type)
    return _sim.s_plus(
        matrix1, matrix2=matrix2,
        l2=1,
        c1=alpha, c2=1-alpha,
        k=k,
        stabilized_shrink=stabilized_shrink,
        bayesian_shrink=bayesian_shrink,
        additive_shrink=additive_shrink,
        threshold=threshold,
        binary=binary,
        target_rows=target_rows,
        target_cols=target_cols,
        filter_cols=filter_cols,
        verbose=verbose,
        format_output=format_output,
        num_threads=num_threads
    )
Jaccard Similarity

Compute Jaccard similarity (intersection over union).

Parameters:

Name Type Description Default
matrix1 spmatrix

Input sparse matrix.

 required
matrix2 Optional[spmatrix]

Optional second matrix. If None, uses matrix1.T.

 None
k int

Number of top-k items per row to keep.

 100
shrink float

Shrinkage value that prevents instability when normalizations are small.

 0.0
shrink_type Literal['stabilized', 'bayesian', 'additive']

Type of shrinkage: 'stabilized', 'bayesian', 'additive'.

 'stabilized'
threshold float

Minimum similarity value to retain.

 0.0
binary bool

Whether to binarize the input matrix before computation.

 False
target_rows Optional[Union[list[int], ndarray]]

List or array of row indices to compute. If None, computes all.

 None
target_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to include before top-k. Can be a list or sparse mask matrix.

 None
filter_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to exclude before top-k. Can be a list or sparse mask matrix.

 None
verbose bool

Whether to show a progress bar.

 True
format_output Literal['csr', 'coo']

Output format, either 'csr' or 'coo'. Use 'coo' on Windows.

 'coo'
num_threads int

Number of threads to use (0 = all available cores).

 0

Returns:

Type Description
spmatrix

A sparse matrix of top-k Jaccard similarities in the specified format.
Source code in similaripy/similarity.py 
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def jaccard(
    matrix1: spmatrix,
    matrix2: Optional[spmatrix] = None,
    k: int = 100,
    shrink: float = 0.0,
    shrink_type: Literal['stabilized', 'bayesian', 'additive'] = 'stabilized',
    threshold: float = 0.0,
    binary: bool = False,
    target_rows: Optional[Union[list[int], np.ndarray]] = None,
    target_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    filter_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    verbose: bool = True,
    format_output: Literal['csr', 'coo'] = 'coo',
    num_threads: int = 0
) -> spmatrix:
    """
    Compute Jaccard similarity (intersection over union).

    Args:
        matrix1: Input sparse matrix.
        matrix2: Optional second matrix. If None, uses matrix1.T.
        k: Number of top-k items per row to keep.
        shrink: Shrinkage value that prevents instability when normalizations are small.
        shrink_type: Type of shrinkage: 'stabilized', 'bayesian', 'additive'.
        threshold: Minimum similarity value to retain.
        binary: Whether to binarize the input matrix before computation.
        target_rows: List or array of row indices to compute. If None, computes all.
        target_cols: Columns to include before top-k. Can be a list or sparse mask matrix.
        filter_cols: Columns to exclude before top-k. Can be a list or sparse mask matrix.
        verbose: Whether to show a progress bar.
        format_output: Output format, either 'csr' or 'coo'. Use 'coo' on Windows.
        num_threads: Number of threads to use (0 = all available cores).

    Returns:
        A sparse matrix of top-k Jaccard similarities in the specified format.
    """
    stabilized_shrink, bayesian_shrink, additive_shrink = __get_shrink_values__(shrink, shrink_type)
    return _sim.s_plus(
        matrix1, matrix2=matrix2,
        l1=1,
        t1=1, t2=1,
        k=k,
        stabilized_shrink=stabilized_shrink,
        bayesian_shrink=bayesian_shrink,
        additive_shrink=additive_shrink,
        threshold=threshold,
        binary=binary,
        target_rows=target_rows,
        target_cols=target_cols,
        filter_cols=filter_cols,
        verbose=verbose,
        format_output=format_output,
        num_threads=num_threads
    )
Dice Similarity

Compute Dice similarity (harmonic mean of overlap and size).

Parameters:

Name Type Description Default
matrix1 spmatrix

Input sparse matrix.

 required
matrix2 Optional[spmatrix]

Optional second matrix. If None, uses matrix1.T.

 None
k int

Number of top-k items per row to keep.

 100
shrink float

Shrinkage value that prevents instability when normalizations are small.

 0.0
shrink_type Literal['stabilized', 'bayesian', 'additive']

Type of shrinkage: 'stabilized', 'bayesian', 'additive'.

 'stabilized'
threshold float

Minimum similarity value to retain.

 0.0
binary bool

Whether to binarize the input matrix before computation.

 False
target_rows Optional[Union[list[int], ndarray]]

List or array of row indices to compute. If None, computes all.

 None
target_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to include before top-k. Can be a list or sparse mask matrix.

 None
filter_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to exclude before top-k. Can be a list or sparse mask matrix.

 None
verbose bool

Whether to show a progress bar.

 True
format_output Literal['csr', 'coo']

Output format, either 'csr' or 'coo'. Use 'coo' on Windows.

 'coo'
num_threads int

Number of threads to use (0 = all available cores).

 0

Returns:

Type Description
spmatrix

A sparse matrix of top-k Dice similarities in the specified format.
Source code in similaripy/similarity.py 
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def dice(
    matrix1: spmatrix,
    matrix2: Optional[spmatrix] = None,
    k: int = 100,
    shrink: float = 0.0,
    shrink_type: Literal['stabilized', 'bayesian', 'additive'] = 'stabilized',
    threshold: float = 0.0,
    binary: bool = False,
    target_rows: Optional[Union[list[int], np.ndarray]] = None,
    target_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    filter_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    verbose: bool = True,
    format_output: Literal['csr', 'coo'] = 'coo',
    num_threads: int = 0
) -> spmatrix:
    """
    Compute Dice similarity (harmonic mean of overlap and size).

    Args:
        matrix1: Input sparse matrix.
        matrix2: Optional second matrix. If None, uses matrix1.T.
        k: Number of top-k items per row to keep.
        shrink: Shrinkage value that prevents instability when normalizations are small.
        shrink_type: Type of shrinkage: 'stabilized', 'bayesian', 'additive'.
        threshold: Minimum similarity value to retain.
        binary: Whether to binarize the input matrix before computation.
        target_rows: List or array of row indices to compute. If None, computes all.
        target_cols: Columns to include before top-k. Can be a list or sparse mask matrix.
        filter_cols: Columns to exclude before top-k. Can be a list or sparse mask matrix.
        verbose: Whether to show a progress bar.
        format_output: Output format, either 'csr' or 'coo'. Use 'coo' on Windows.
        num_threads: Number of threads to use (0 = all available cores).

    Returns:
        A sparse matrix of top-k Dice similarities in the specified format.
    """
    stabilized_shrink, bayesian_shrink, additive_shrink = __get_shrink_values__(shrink, shrink_type)
    return _sim.s_plus(
        matrix1, matrix2=matrix2,
        l1=1,
        t1=0.5, t2=0.5,
        k=k,
        stabilized_shrink=stabilized_shrink,
        bayesian_shrink=bayesian_shrink,
        additive_shrink=additive_shrink,
        threshold=threshold,
        binary=binary,
        target_rows=target_rows,
        target_cols=target_cols,
        filter_cols=filter_cols,
        verbose=verbose,
        format_output=format_output,
        num_threads=num_threads
    )
Tversky Similarity

Compute Tversky similarity between sparse vectors.

Parameters:

Name Type Description Default
matrix1 spmatrix

Input sparse matrix.

 required
matrix2 Optional[spmatrix]

Optional second matrix. If None, uses matrix1.T.

 None
alpha float

Tversky weight for elements unique to matrix1.

 1.0
beta float

Tversky weight for elements unique to matrix2.

 1.0
k int

Number of top-k items per row to keep.

 100
shrink float

Shrinkage value that prevents instability when normalizations are small.

 0.0
shrink_type Literal['stabilized', 'bayesian', 'additive']

Type of shrinkage: 'stabilized', 'bayesian', 'additive'.

 'stabilized'
threshold float

Minimum similarity value to retain.

 0.0
binary bool

Whether to binarize the input matrix before computation.

 False
target_rows Optional[Union[list[int], ndarray]]

List or array of row indices to compute. If None, computes all.

 None
target_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to include before top-k. Can be a list or sparse mask matrix.

 None
filter_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to exclude before top-k. Can be a list or sparse mask matrix.

 None
verbose bool

Whether to show a progress bar.

 True
format_output Literal['csr', 'coo']

Output format, either 'csr' or 'coo'. Use 'coo' on Windows.

 'coo'
num_threads int

Number of threads to use (0 = all available cores).

 0

Returns:

Type Description
spmatrix

A sparse matrix of top-k Tversky similarities in the specified format.
Source code in similaripy/similarity.py 
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def tversky(
    matrix1: spmatrix,
    matrix2: Optional[spmatrix] = None,
    alpha: float = 1.0,
    beta: float = 1.0,
    k: int = 100,
    shrink: float = 0.0,
    shrink_type: Literal['stabilized', 'bayesian', 'additive'] = 'stabilized',
    threshold: float = 0.0,
    binary: bool = False,
    target_rows: Optional[Union[list[int], np.ndarray]] = None,
    target_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    filter_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    verbose: bool = True,
    format_output: Literal['csr', 'coo'] = 'coo',
    num_threads: int = 0
) -> spmatrix:
    """
    Compute Tversky similarity between sparse vectors.

    Args:
        matrix1: Input sparse matrix.
        matrix2: Optional second matrix. If None, uses matrix1.T.
        alpha: Tversky weight for elements unique to matrix1.
        beta: Tversky weight for elements unique to matrix2.
        k: Number of top-k items per row to keep.
        shrink: Shrinkage value that prevents instability when normalizations are small.
        shrink_type: Type of shrinkage: 'stabilized', 'bayesian', 'additive'.
        threshold: Minimum similarity value to retain.
        binary: Whether to binarize the input matrix before computation.
        target_rows: List or array of row indices to compute. If None, computes all.
        target_cols: Columns to include before top-k. Can be a list or sparse mask matrix.
        filter_cols: Columns to exclude before top-k. Can be a list or sparse mask matrix.
        verbose: Whether to show a progress bar.
        format_output: Output format, either 'csr' or 'coo'. Use 'coo' on Windows.
        num_threads: Number of threads to use (0 = all available cores).

    Returns:
        A sparse matrix of top-k Tversky similarities in the specified format.
    """
    stabilized_shrink, bayesian_shrink, additive_shrink = __get_shrink_values__(shrink, shrink_type)
    return _sim.s_plus(
        matrix1, matrix2=matrix2,
        l1=1,
        t1=alpha, t2=beta,
        k=k,
        stabilized_shrink=stabilized_shrink,
        bayesian_shrink=bayesian_shrink,
        additive_shrink=additive_shrink,
        threshold=threshold,
        binary=binary,
        target_rows=target_rows,
        target_cols=target_cols,
        filter_cols=filter_cols,
        verbose=verbose,
        format_output=format_output,
        num_threads=num_threads
    )

Graph-Based Similarities

P3α

Compute P3alpha similarity using a normalized 3-step random walk.

Parameters:

Name Type Description Default
matrix1 spmatrix

Input sparse matrix.

 required
matrix2 Optional[spmatrix]

Optional second matrix. If None, uses matrix1.T.

 None
alpha float

Exponent for transition probabilities to control popularity effect.

 1.0
k int

Number of top-k items per row to keep.

 100
shrink float

Shrinkage value that prevents instability when normalizations are small.

 0.0
shrink_type Literal['stabilized', 'bayesian', 'additive']

Type of shrinkage: 'stabilized', 'bayesian', 'additive'.

 'stabilized'
threshold float

Minimum similarity value to retain.

 0.0
binary bool

Whether to binarize the input matrix before computation.

 False
target_rows Optional[Union[list[int], ndarray]]

List or array of row indices to compute. If None, computes all.

 None
target_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to include before top-k. Can be a list or sparse mask matrix.

 None
filter_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to exclude before top-k. Can be a list or sparse mask matrix.

 None
verbose bool

Whether to show a progress bar.

 True
format_output Literal['csr', 'coo']

Output format, either 'csr' or 'coo'. Use 'coo' on Windows.

 'coo'
num_threads int

Number of threads to use (0 = all available cores).

 0

Returns:

Type Description
spmatrix

A sparse matrix of top-k P3alpha similarities in the specified format.
Source code in similaripy/similarity.py 
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def p3alpha(
    matrix1: spmatrix,
    matrix2: Optional[spmatrix] = None,
    alpha: float = 1.0,
    k: int = 100,
    shrink: float = 0.0,
    shrink_type: Literal['stabilized', 'bayesian', 'additive'] = 'stabilized',
    threshold: float = 0.0,
    binary: bool = False,
    target_rows: Optional[Union[list[int], np.ndarray]] = None,
    target_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    filter_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    verbose: bool = True,
    format_output: Literal['csr', 'coo'] = 'coo',
    num_threads: int = 0
) -> spmatrix:
    """
    Compute P3alpha similarity using a normalized 3-step random walk.

    Args:
        matrix1: Input sparse matrix.
        matrix2: Optional second matrix. If None, uses matrix1.T.
        alpha: Exponent for transition probabilities to control popularity effect.
        k: Number of top-k items per row to keep.
        shrink: Shrinkage value that prevents instability when normalizations are small.
        shrink_type: Type of shrinkage: 'stabilized', 'bayesian', 'additive'.
        threshold: Minimum similarity value to retain.
        binary: Whether to binarize the input matrix before computation.
        target_rows: List or array of row indices to compute. If None, computes all.
        target_cols: Columns to include before top-k. Can be a list or sparse mask matrix.
        filter_cols: Columns to exclude before top-k. Can be a list or sparse mask matrix.
        verbose: Whether to show a progress bar.
        format_output: Output format, either 'csr' or 'coo'. Use 'coo' on Windows.
        num_threads: Number of threads to use (0 = all available cores).

    Returns:
        A sparse matrix of top-k P3alpha similarities in the specified format.
    """
    if matrix2 is None:
        matrix2 = matrix1.T
    matrix1 = _normalize(matrix1, norm='l1', axis=1, inplace=False)
    matrix1.data = np.power(matrix1.data, alpha)
    matrix2 = _normalize(matrix2, norm='l1', axis=1, inplace=False)
    matrix2.data = np.power(matrix2.data, alpha)
    stabilized_shrink, bayesian_shrink, additive_shrink = __get_shrink_values__(shrink, shrink_type)
    return _sim.s_plus(
        matrix1=matrix1, matrix2=matrix2,
        k=k,
        stabilized_shrink=stabilized_shrink,
        bayesian_shrink=bayesian_shrink,
        additive_shrink=additive_shrink,
        threshold=threshold,
        binary=binary,
        target_rows=target_rows,
        target_cols=target_cols,
        filter_cols=filter_cols,
        verbose=verbose,
        format_output=format_output,
        num_threads=num_threads
    )
RP3β

Compute RP3beta similarity: P3alpha with popularity penalization.

Parameters:

Name Type Description Default
matrix1 spmatrix

Input sparse matrix.

 required
matrix2 Optional[spmatrix]

Optional second matrix. If None, uses matrix1.T.

 None
alpha float

Exponent for transition probabilities.

 1.0
beta float

Exponent to penalize popularity based on column sums.

 1.0
k int

Number of top-k items per row to keep.

 100
shrink float

Shrinkage value that prevents instability when normalizations are small.

 0.0
shrink_type Literal['stabilized', 'bayesian', 'additive']

Type of shrinkage: 'stabilized', 'bayesian', 'additive'.

 'stabilized'
threshold float

Minimum similarity value to retain.

 0.0
binary bool

Whether to binarize the input matrix before computation.

 False
target_rows Optional[Union[list[int], ndarray]]

List or array of row indices to compute. If None, computes all.

 None
target_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to include before top-k. Can be a list or sparse mask matrix.

 None
filter_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to exclude before top-k. Can be a list or sparse mask matrix.

 None
verbose bool

Whether to show a progress bar.

 True
format_output Literal['csr', 'coo']

Output format, either 'csr' or 'coo'. Use 'coo' on Windows.

 'coo'
num_threads int

Number of threads to use (0 = all available cores).

 0

Returns:

Type Description
spmatrix

A sparse matrix of top-k RP3beta similarities in the specified format.
Source code in similaripy/similarity.py 
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def rp3beta(
    matrix1: spmatrix,
    matrix2: Optional[spmatrix] = None,
    alpha: float = 1.0,
    beta: float = 1.0,
    k: int = 100,
    shrink: float = 0.0,
    shrink_type: Literal['stabilized', 'bayesian', 'additive'] = 'stabilized',
    threshold: float = 0.0,
    binary: bool = False,
    target_rows: Optional[Union[list[int], np.ndarray]] = None,
    target_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    filter_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    verbose: bool = True,
    format_output: Literal['csr', 'coo'] = 'coo',
    num_threads: int = 0
) -> spmatrix:
    """
    Compute RP3beta similarity: P3alpha with popularity penalization.

    Args:
        matrix1: Input sparse matrix.
        matrix2: Optional second matrix. If None, uses matrix1.T.
        alpha: Exponent for transition probabilities.
        beta: Exponent to penalize popularity based on column sums.
        k: Number of top-k items per row to keep.
        shrink: Shrinkage value that prevents instability when normalizations are small.
        shrink_type: Type of shrinkage: 'stabilized', 'bayesian', 'additive'.
        threshold: Minimum similarity value to retain.
        binary: Whether to binarize the input matrix before computation.
        target_rows: List or array of row indices to compute. If None, computes all.
        target_cols: Columns to include before top-k. Can be a list or sparse mask matrix.
        filter_cols: Columns to exclude before top-k. Can be a list or sparse mask matrix.
        verbose: Whether to show a progress bar.
        format_output: Output format, either 'csr' or 'coo'. Use 'coo' on Windows.
        num_threads: Number of threads to use (0 = all available cores).

    Returns:
        A sparse matrix of top-k RP3beta similarities in the specified format.
    """
    if matrix2 is None:
        matrix2 = matrix1.T
    pop_m2 = matrix2.sum(axis=0).A1
    matrix1 = _normalize(matrix1, norm='l1', axis=1, inplace=False)
    matrix1.data = np.power(matrix1.data, alpha)
    matrix2 = _normalize(matrix2, norm='l1', axis=1, inplace=False)
    matrix2.data = np.power(matrix2.data, alpha)
    stabilized_shrink, bayesian_shrink, additive_shrink = __get_shrink_values__(shrink, shrink_type)
    return _sim.s_plus(
        matrix1=matrix1, matrix2=matrix2,
        weight_depop_matrix2=pop_m2,
        p2=beta,
        l3=1,
        k=k,
        stabilized_shrink=stabilized_shrink,
        bayesian_shrink=bayesian_shrink,
        additive_shrink=additive_shrink,
        threshold=threshold,
        binary=binary,
        target_rows=target_rows,
        target_cols=target_cols,
        filter_cols=filter_cols,
        verbose=verbose,
        format_output=format_output,
        num_threads=num_threads
    )

Hybrid Similarity

S Plus

Combines Tversky and Cosine normalizations with RP3Beta-style depopularization, fully controlled by tunable weights

Parameters:

Name Type Description Default
matrix1 spmatrix

Input sparse matrix.

 required
matrix2 Optional[spmatrix]

Optional second matrix. If None, uses matrix1.T.

 None
l1 float

Tversky normalization strength.

 0.5
l2 float

Cosine normalization strength.

 0.5
l3 float

Popularity penalization strength.

 0.0
t1 float

Tversky alpha for matrix1.

 1.0
t2 float

Tversky beta for matrix2.

 1.0
c1 float

Cosine exponent coefficient for matrix1.

 0.5
c2 float

Cosine exponent coefficient for matrix2.

 0.5
pop1 Optional[Union[Literal['none', 'sum'], ndarray]]

Popularity weights for matrix1. 'none', 'sum', or custom array.

 'none'
pop2 Optional[Union[Literal['none', 'sum'], ndarray]]

Popularity weights for matrix2. 'none', 'sum', or custom array.

 'none'
alpha float

Coefficient applied on the raw similarity value before normalizations.

 1.0
beta1 float

Popularity penalization coefficient for matrix1 items.

 0.0
beta2 float

Popularity penalization coefficient for matrix2 items.

 0.0
k int

Number of top-k items per row to keep.

 100
shrink float

Shrinkage value that prevents instability when normalizations are small.

 0.0
shrink_type Literal['stabilized', 'bayesian', 'additive']

Type of shrinkage: 'stabilized', 'bayesian', 'additive'.

 'stabilized'
threshold float

Minimum similarity value to retain.

 0.0
binary bool

Whether to binarize the input matrix before computation.

 False
target_rows Optional[Union[list[int], ndarray]]

List or array of row indices to compute. If None, computes all.

 None
target_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to include before top-k. Can be a list or sparse mask matrix.

 None
filter_cols Optional[Union[list[int], ndarray, spmatrix]]

Columns to exclude before top-k. Can be a list or sparse mask matrix.

 None
verbose bool

Whether to show a progress bar.

 True
format_output Literal['csr', 'coo']

Output format, either 'csr' or 'coo'. Use 'coo' on Windows.

 'coo'
num_threads int

Number of threads to use (0 = all available cores).

 0

Returns:

Type Description
spmatrix

A sparse matrix of top-k s_plus similarities in the specified format.
Source code in similaripy/similarity.py 
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def s_plus(
    matrix1: spmatrix,
    matrix2: Optional[spmatrix] = None,
    l1: float = 0.5,
    l2: float = 0.5,
    l3: float = 0.0,
    t1: float = 1.0,
    t2: float = 1.0,
    c1: float = 0.5,
    c2: float = 0.5,
    pop1: Optional[Union[Literal['none','sum'], np.ndarray]]= 'none',
    pop2: Optional[Union[Literal['none','sum'], np.ndarray]]= 'none',
    alpha: float = 1.0,
    beta1: float = 0.0,
    beta2: float = 0.0,
    k: int = 100,
    shrink: float = 0.0,
    shrink_type: Literal['stabilized', 'bayesian', 'additive'] = 'stabilized',
    threshold: float = 0.0,
    binary: bool = False,
    target_rows: Optional[Union[list[int], np.ndarray]] = None,
    target_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    filter_cols: Optional[Union[list[int], np.ndarray, spmatrix]] = None,
    verbose: bool = True,
    format_output: Literal['csr', 'coo'] = 'coo',
    num_threads: int = 0
) -> spmatrix:
    """
    Combines Tversky and Cosine normalizations with RP3Beta-style depopularization, fully controlled by tunable weights

    Args:
        matrix1: Input sparse matrix.
        matrix2: Optional second matrix. If None, uses matrix1.T.
        l1: Tversky normalization strength.
        l2: Cosine normalization strength.
        l3: Popularity penalization strength.
        t1: Tversky alpha for matrix1.
        t2: Tversky beta for matrix2.
        c1: Cosine exponent coefficient for matrix1.
        c2: Cosine exponent coefficient for matrix2.
        pop1: Popularity weights for matrix1. 'none', 'sum', or custom array.
        pop2: Popularity weights for matrix2. 'none', 'sum', or custom array.
        alpha: Coefficient applied on the raw similarity value before normalizations.
        beta1: Popularity penalization coefficient for matrix1 items.
        beta2: Popularity penalization coefficient for matrix2 items.
        k: Number of top-k items per row to keep.
        shrink: Shrinkage value that prevents instability when normalizations are small.
        shrink_type: Type of shrinkage: 'stabilized', 'bayesian', 'additive'.
        threshold: Minimum similarity value to retain.
        binary: Whether to binarize the input matrix before computation.
        target_rows: List or array of row indices to compute. If None, computes all.
        target_cols: Columns to include before top-k. Can be a list or sparse mask matrix.
        filter_cols: Columns to exclude before top-k. Can be a list or sparse mask matrix.
        verbose: Whether to show a progress bar.
        format_output: Output format, either 'csr' or 'coo'. Use 'coo' on Windows.
        num_threads: Number of threads to use (0 = all available cores).

    Returns:
        A sparse matrix of top-k s_plus similarities in the specified format.
    """
    stabilized_shrink, bayesian_shrink, additive_shrink = __get_shrink_values__(shrink, shrink_type)
    return _sim.s_plus(
        matrix1, matrix2=matrix2,
        l1=l1, l2=l2, l3=l3,
        t1=t1, t2=t2,
        c1=c1, c2=c2,
        a1=alpha,
        weight_depop_matrix1=pop1,
        weight_depop_matrix2=pop2,
        p1=beta1,
        p2=beta2,
        k=k,
        stabilized_shrink=stabilized_shrink,
        bayesian_shrink=bayesian_shrink,
        additive_shrink=additive_shrink,
        threshold=threshold,
        binary=binary,
        target_rows=target_rows,
        target_cols=target_cols,
        filter_cols=filter_cols,
        verbose=verbose,
        format_output=format_output,
        num_threads=num_threads
    )