Correlation via fft implementation (#2203)

* Added keyword `method` to `correlate` function similar to [scipy
correlate](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.correlate.html)
* If `method == 'auto'` method is choosing automatically between
`direct` and `fft`
* Added implementation of fft-based correlation
* fft-based implementation may have accuracy issues, so it is validated
in non-standard way.

Depends on: #2180,
#2202

---------

Co-authored-by: Anton <100830759+antonwolfy@users.noreply.github.com>

Author: AlexanderKalistratov

dpnp/dpnp_iface_statistics.py

@@ -37,6 +37,8 @@
 
 """
 
+import math
+
 import dpctl.tensor as dpt
 import dpctl.tensor._tensor_elementwise_impl as ti
 import dpctl.utils as dpu
@@ -481,24 +483,66 @@ def _get_padding(a_size, v_size, mode):
         r_pad = v_size - l_pad - 1
     elif mode == "full":
         l_pad, r_pad = v_size - 1, v_size - 1
-    else:
+    else:  # pragma: no cover
         raise ValueError(
             f"Unknown mode: {mode}. Only 'valid', 'same', 'full' are supported."
         )
 
     return l_pad, r_pad
 
 
-def _run_native_sliding_dot_product1d(a, v, l_pad, r_pad):
+def _choose_conv_method(a, v, rdtype):
+    assert a.size >= v.size
+    if rdtype == dpnp.bool:
+        # to avoid accuracy issues
+        return "direct"
+
+    if v.size < 10**4 or a.size < 10**4:
+        # direct method is faster for small arrays
+        return "direct"
+
+    if dpnp.issubdtype(rdtype, dpnp.integer):
+        max_a = int(dpnp.max(dpnp.abs(a)))
+        sum_v = int(dpnp.sum(dpnp.abs(v)))
+        max_value = int(max_a * sum_v)
+
+        default_float = dpnp.default_float_type(a.sycl_device)
+        if max_value > 2 ** numpy.finfo(default_float).nmant - 1:
+            # can't represent the result in the default float type
+            return "direct"  # pragma: no covers
+
+    if dpnp.issubdtype(rdtype, dpnp.number):
+        return "fft"
+
+    raise ValueError(f"Unsupported dtype: {rdtype}")  # pragma: no cover
+
+
+def _run_native_sliding_dot_product1d(a, v, l_pad, r_pad, rdtype):
     queue = a.sycl_queue
+    device = a.sycl_device
+
+    supported_types = statistics_ext.sliding_dot_product1d_dtypes()
+    supported_dtype = to_supported_dtypes(rdtype, supported_types, device)
+
+    if supported_dtype is None:  # pragma: no cover
+        raise ValueError(
+            f"function does not support input types "
+            f"({a.dtype.name}, {v.dtype.name}), "
+            "and the inputs could not be coerced to any "
+            f"supported types. List of supported types: "
+            f"{[st.name for st in supported_types]}"
+        )
+
+    a_casted = dpnp.asarray(a, dtype=supported_dtype, order="C")
+    v_casted = dpnp.asarray(v, dtype=supported_dtype, order="C")
 
-    usm_type = dpu.get_coerced_usm_type([a.usm_type, v.usm_type])
-    out_size = l_pad + r_pad + a.size - v.size + 1
+    usm_type = dpu.get_coerced_usm_type([a_casted.usm_type, v_casted.usm_type])
+    out_size = l_pad + r_pad + a_casted.size - v_casted.size + 1
     # out type is the same as input type
-    out = dpnp.empty_like(a, shape=out_size, usm_type=usm_type)
+    out = dpnp.empty_like(a_casted, shape=out_size, usm_type=usm_type)
 
-    a_usm = dpnp.get_usm_ndarray(a)
-    v_usm = dpnp.get_usm_ndarray(v)
+    a_usm = dpnp.get_usm_ndarray(a_casted)
+    v_usm = dpnp.get_usm_ndarray(v_casted)
     out_usm = dpnp.get_usm_ndarray(out)
 
     _manager = dpu.SequentialOrderManager[queue]
@@ -516,7 +560,30 @@ def _run_native_sliding_dot_product1d(a, v, l_pad, r_pad):
     return out
 
 
-def correlate(a, v, mode="valid"):
+def _convolve_fft(a, v, l_pad, r_pad, rtype):
+    assert a.size >= v.size
+    assert l_pad < v.size
+
+    # +1 is needed to avoid circular convolution
+    padded_size = a.size + r_pad + 1
+    fft_size = 2 ** int(math.ceil(math.log2(padded_size)))
+
+    af = dpnp.fft.fft(a, fft_size)  # pylint: disable=no-member
+    vf = dpnp.fft.fft(v, fft_size)  # pylint: disable=no-member
+
+    r = dpnp.fft.ifft(af * vf)  # pylint: disable=no-member
+    if dpnp.issubdtype(rtype, dpnp.floating):
+        r = r.real
+    elif dpnp.issubdtype(rtype, dpnp.integer) or rtype == dpnp.bool:
+        r = r.real.round()
+
+    start = v.size - 1 - l_pad
+    end = padded_size - 1
+
+    return r[start:end]
+
+
+def correlate(a, v, mode="valid", method="auto"):
     r"""
     Cross-correlation of two 1-dimensional sequences.
 
@@ -541,6 +608,20 @@ def correlate(a, v, mode="valid"):
         is ``"valid"``, unlike :obj:`dpnp.convolve`, which uses ``"full"``.
 
         Default: ``"valid"``.
+    method : {"auto", "direct", "fft"}, optional
+        Specifies which method to use to calculate the correlation:
+
+        - `"direct"` : The correlation is determined directly from sums.
+        - `"fft"` : The Fourier Transform is used to perform the calculations.
+          This method is faster for long sequences but can have accuracy issues.
+        - `"auto"` : Automatically chooses direct or Fourier method based on
+          an estimate of which is faster.
+
+        Note: Use of the FFT convolution on input containing NAN or INF
+        will lead to the entire output being NAN or INF.
+        Use method='direct' when your input contains NAN or INF values.
+
+        Default: ``"auto"``.
 
     Returns
     -------
@@ -608,20 +689,14 @@ def correlate(a, v, mode="valid"):
             f"Received shapes: a.shape={a.shape}, v.shape={v.shape}"
         )
 
-    supported_types = statistics_ext.sliding_dot_product1d_dtypes()
+    supported_methods = ["auto", "direct", "fft"]
+    if method not in supported_methods:
+        raise ValueError(
+            f"Unknown method: {method}. Supported methods: {supported_methods}"
+        )
 
     device = a.sycl_device
     rdtype = result_type_for_device([a.dtype, v.dtype], device)
-    supported_dtype = to_supported_dtypes(rdtype, supported_types, device)
-
-    if supported_dtype is None:  # pragma: no cover
-        raise ValueError(
-            f"function does not support input types "
-            f"({a.dtype.name}, {v.dtype.name}), "
-            "and the inputs could not be coerced to any "
-            f"supported types. List of supported types: "
-            f"{[st.name for st in supported_types]}"
-        )
 
     if dpnp.issubdtype(v.dtype, dpnp.complexfloating):
         v = dpnp.conj(v)
@@ -633,10 +708,15 @@ def correlate(a, v, mode="valid"):
 
     l_pad, r_pad = _get_padding(a.size, v.size, mode)
 
-    a_casted = dpnp.asarray(a, dtype=supported_dtype, order="C")
-    v_casted = dpnp.asarray(v, dtype=supported_dtype, order="C")
+    if method == "auto":
+        method = _choose_conv_method(a, v, rdtype)
 
-    r = _run_native_sliding_dot_product1d(a_casted, v_casted, l_pad, r_pad)
+    if method == "direct":
+        r = _run_native_sliding_dot_product1d(a, v, l_pad, r_pad, rdtype)
+    elif method == "fft":
+        r = _convolve_fft(a, v[::-1], l_pad, r_pad, rdtype)
+    else:  # pragma: no cover
+        raise ValueError(f"Unknown method: {method}")
 
     if revert:
         r = r[::-1]

dpnp/tests/helper.py

@@ -13,6 +13,7 @@ def assert_dtype_allclose(
     check_type=True,
     check_only_type_kind=False,
     factor=8,
+    relative_factor=None,
 ):
     """
     Assert DPNP and NumPy array based on maximum dtype resolution of input arrays
@@ -183,6 +184,7 @@ def generate_random_numpy_array(
     seed_value=None,
     low=-10,
     high=10,
+    probability=0.5,
 ):
     """
     Generate a random numpy array with the specified shape and dtype.
@@ -197,23 +199,32 @@ def generate_random_numpy_array(
     dtype : str or dtype, optional
         Desired data-type for the output array.
         If not specified, data type will be determined by numpy.
+
         Default : ``None``
     order : {"C", "F"}, optional
         Specify the memory layout of the output array.
+
         Default: ``"C"``.
     hermitian : bool, optional
         If True, generates a Hermitian (symmetric if `dtype` is real) matrix.
+
         Default : ``False``
     seed_value : int, optional
         The seed value to initialize the random number generator.
+
         Default : ``None``
     low : {int, float}, optional
         Lower boundary of the generated samples from a uniform distribution.
+
         Default : ``-10``.
     high : {int, float}, optional
         Upper boundary of the generated samples from a uniform distribution.
+
         Default : ``10``.
+    probability : float, optional
+        If dtype is bool, the probability of True. Ignored for other dtypes.
 
+        Default : ``0.5``.
     Returns
     -------
     out : numpy.ndarray
@@ -232,9 +243,15 @@ def generate_random_numpy_array(
 
     # dtype=int is needed for 0d arrays
     size = numpy.prod(shape, dtype=int)
-    a = numpy.random.uniform(low, high, size).astype(dtype)
-    if numpy.issubdtype(a.dtype, numpy.complexfloating):
-        a += 1j * numpy.random.uniform(low, high, size)
+    if dtype == dpnp.bool:
+        a = numpy.random.choice(
+            [False, True], size, p=[1 - probability, probability]
+        )
+    else:
+        a = numpy.random.uniform(low, high, size).astype(dtype)
+
+        if numpy.issubdtype(a.dtype, numpy.complexfloating):
+            a += 1j * numpy.random.uniform(low, high, size)
 
     a = a.reshape(shape)
     if hermitian and a.size > 0:

dpnp/tests/test_statistics.py

@@ -180,26 +180,101 @@ def test_corrcoef_scalar(self):
 
 
 class TestCorrelate:
+    @staticmethod
+    def _get_kwargs(mode=None, method=None):
+        dpnp_kwargs = {}
+        numpy_kwargs = {}
+        if mode is not None:
+            dpnp_kwargs["mode"] = mode
+            numpy_kwargs["mode"] = mode
+        if method is not None:
+            dpnp_kwargs["method"] = method
+        return dpnp_kwargs, numpy_kwargs
+
+    def setup_method(self):
+        numpy.random.seed(0)
+
     @pytest.mark.parametrize(
         "a, v", [([1], [1, 2, 3]), ([1, 2, 3], [1]), ([1, 2, 3], [1, 2])]
     )
     @pytest.mark.parametrize("mode", [None, "full", "valid", "same"])
     @pytest.mark.parametrize("dtype", get_all_dtypes(no_bool=True))
-    def test_correlate(self, a, v, mode, dtype):
+    @pytest.mark.parametrize("method", [None, "auto", "direct", "fft"])
+    def test_correlate(self, a, v, mode, dtype, method):
         an = numpy.array(a, dtype=dtype)
         vn = numpy.array(v, dtype=dtype)
         ad = dpnp.array(an)
         vd = dpnp.array(vn)
 
-        if mode is None:
-            expected = numpy.correlate(an, vn)
-            result = dpnp.correlate(ad, vd)
-        else:
-            expected = numpy.correlate(an, vn, mode=mode)
-            result = dpnp.correlate(ad, vd, mode=mode)
+        dpnp_kwargs, numpy_kwargs = self._get_kwargs(mode, method)
+
+        expected = numpy.correlate(an, vn, **numpy_kwargs)
+        result = dpnp.correlate(ad, vd, **dpnp_kwargs)
 
         assert_dtype_allclose(result, expected)
 
+    @pytest.mark.parametrize("a_size", [1, 100, 10000])
+    @pytest.mark.parametrize("v_size", [1, 100, 10000])
+    @pytest.mark.parametrize("mode", ["full", "valid", "same"])
+    @pytest.mark.parametrize("dtype", get_all_dtypes(no_none=True))
+    @pytest.mark.parametrize("method", ["auto", "direct", "fft"])
+    def test_correlate_random(self, a_size, v_size, mode, dtype, method):
+        an = generate_random_numpy_array(a_size, dtype, probability=0.9)
+        vn = generate_random_numpy_array(v_size, dtype, probability=0.9)
+
+        ad = dpnp.array(an)
+        vd = dpnp.array(vn)
+
+        dpnp_kwargs, numpy_kwargs = self._get_kwargs(mode, method)
+
+        result = dpnp.correlate(ad, vd, **dpnp_kwargs)
+        expected = numpy.correlate(an, vn, **numpy_kwargs)
+
+        rdtype = result.dtype
+        if dpnp.issubdtype(rdtype, dpnp.integer):
+            rdtype = dpnp.default_float_type(ad.device)
+
+        if method != "fft" and (
+            dpnp.issubdtype(dtype, dpnp.integer) or dtype == dpnp.bool
+        ):
+            # For 'direct' and 'auto' methods, we expect exact results for integer types
+            assert_array_equal(result, expected)
+        else:
+            result = result.astype(rdtype)
+            if method == "direct":
+                expected = numpy.correlate(an, vn, **numpy_kwargs)
+                # For 'direct' method we can use standard validation
+                # acceptable error depends on the kernel size
+                # while error grows linearly with the kernel size,
+                # this empirically found formula provides a good balance
+                # the resulting factor is 40 for kernel size = 1,
+                # 400 for kernel size = 100 and 4000 for kernel size = 10000
+                factor = int(40 * (min(a_size, v_size) ** 0.5))
+                assert_dtype_allclose(result, expected, factor=factor)
+            else:
+                rtol = 1e-3
+                atol = 1e-3
+
+                if rdtype == dpnp.float64 or rdtype == dpnp.complex128:
+                    rtol = 1e-6
+                    atol = 1e-6
+                elif rdtype == dpnp.bool:
+                    result = result.astype(dpnp.int32)
+                    rdtype = result.dtype
+
+                expected = expected.astype(rdtype)
+
+                diff = numpy.abs(result.asnumpy() - expected)
+                invalid = diff > atol + rtol * numpy.abs(expected)
+
+                # When using the 'fft' method, we might encounter outliers.
+                # This usually happens when the resulting array contains values close to zero.
+                # For these outliers, the relative error can be significant.
+                # We can tolerate a few such outliers.
+                max_outliers = 10 if expected.size > 1 else 0
+                if invalid.sum() > max_outliers:
+                    assert_dtype_allclose(result, expected, factor=1000)
+
     def test_correlate_mode_error(self):
         a = dpnp.arange(5)
         v = dpnp.arange(3)
@@ -240,7 +315,7 @@ def test_correlate_different_sizes(self, size):
         vd = dpnp.array(v)
 
         expected = numpy.correlate(a, v)
-        result = dpnp.correlate(ad, vd)
+        result = dpnp.correlate(ad, vd, method="direct")
 
         assert_dtype_allclose(result, expected, factor=20)
 
@@ -251,6 +326,13 @@ def test_correlate_another_sycl_queue(self):
         with pytest.raises(ValueError):
             dpnp.correlate(a, v)
 
+    def test_correlate_unkown_method(self):
+        a = dpnp.arange(5)
+        v = dpnp.arange(3)
+
+        with pytest.raises(ValueError):
+            dpnp.correlate(a, v, method="unknown")
+
 
 class TestCov:
     @pytest.mark.parametrize(