some style and API consistency updates to linalg norm

docs/src/index.rst

@@ -57,6 +57,7 @@ are the CPU and GPU.
    python/random
    python/transforms
    python/fft
+   python/linalg
    python/nn
    python/optimizers
    python/tree_utils

docs/src/python/linalg.rst

@@ -1,7 +1,7 @@
 .. _linalg:
 
 Linear Algebra
-=====
+==============
 
 .. currentmodule:: mlx.core.linalg
 

mlx/linalg.cpp

@@ -1,47 +1,42 @@
 // Copyright © 2023 Apple Inc.
 
-#include <sstream>
+#include <numeric>
-#include <string>
+#include <ostream>
 #include <vector>
 
 #include "mlx/array.h"
 #include "mlx/linalg.h"
 #include "mlx/ops.h"
-#include "mlx/utils.h"
 
 namespace mlx::core::linalg {
 
+Dtype at_least_float(const Dtype& d) {
+  return is_floating_point(d) ? d : promote_types(d, float32);
+}
+
 inline array vector_norm(
     const array& a,
     const double ord,
     const std::vector<int>& axis,
     bool keepdims,
     StreamOrDevice s) {
-  if (ord == 0.0)
+  auto dtype = at_least_float(a.dtype());
-    return sum(a != 0, axis, keepdims, s);
+  if (ord == 0.0) {
-  else if (ord == 1.0)
+    return astype(sum(not_equal(a, array(0), s), axis, keepdims, s), dtype, s);
-    return sum(abs(a, s), axis, keepdims, s);
+  } else if (ord == 1.0) {
-  else if (ord == 2.0)
+    return astype(sum(abs(a, s), axis, keepdims, s), dtype, s);
-    return sqrt(sum(abs(a, s) * abs(a, s), axis, keepdims, s));
+  } else if (ord == 2.0) {
-  else
+    return sqrt(sum(square(a, s), axis, keepdims, s), s);
+  } else if (ord == std::numeric_limits<double>::infinity()) {
+    return astype(max(abs(a, s), axis, keepdims, s), dtype, s);
+  } else if (ord == -std::numeric_limits<double>::infinity()) {
+    return astype(min(abs(a, s), axis, keepdims, s), dtype, s);
+  } else {
     return power(
-        sum(power(abs(a, s), array(ord), s), axis, keepdims, s),
+        sum(power(abs(a, s), array(ord, dtype), s), axis, keepdims, s),
-        array(1.0 / ord));
+        array(1.0 / ord, dtype),
-}
+        s);
-
+  }
-inline array vector_norm(
-    const array& a,
-    const std::string& ord,
-    const std::vector<int>& axis,
-    bool keepdims,
-    StreamOrDevice s) {
-  if (ord == "inf")
-    return max(abs(a, s), axis, keepdims, s);
-  else if (ord == "-inf")
-    return min(abs(a, s), axis, keepdims, s);
-  std::ostringstream error_stream;
-  error_stream << "Invalid ord value " << ord;
-  throw std::invalid_argument(error_stream.str());
 }
 
 inline array matrix_norm(
@@ -50,19 +45,30 @@ inline array matrix_norm(
     const std::vector<int>& axis,
     bool keepdims,
     StreamOrDevice s) {
+  auto dtype = at_least_float(a.dtype());
   auto row_axis = axis[0];
   auto col_axis = axis[1];
-  if (!keepdims && col_axis > row_axis)
+  if (!keepdims && col_axis > row_axis && col_axis > 0) {
     col_axis -= 1;
-  if (ord == -1.0)
+  }
-    return min(sum(abs(a, s), row_axis, keepdims, s), col_axis, keepdims, s);
+  if (ord == -1.0) {
-  if (ord == 1.0)
+    return astype(
-    return max(sum(abs(a, s), row_axis, keepdims, s), col_axis, keepdims, s);
+        min(sum(abs(a, s), row_axis, keepdims, s), col_axis, keepdims, s),
-  if (ord == 2.0 || ord == -2.0)
+        dtype,
-    throw std::logic_error("Singular value norms are not implemented.");
+        s);
-  std::ostringstream error_stream;
+  } else if (ord == 1.0) {
-  error_stream << "Invalid ord value " << ord << " for matrix norm";
+    return astype(
-  throw std::invalid_argument(error_stream.str());
+        max(sum(abs(a, s), row_axis, keepdims, s), col_axis, keepdims, s),
+        dtype,
+        s);
+  } else if (ord == 2.0 || ord == -2.0) {
+    throw std::runtime_error(
+        "[linalg::norm] Singular value norms are not implemented.");
+  } else {
+    std::ostringstream msg;
+    msg << "[linalg::norm] Invalid ord value " << ord << " for matrix norm";
+    throw std::invalid_argument(msg.str());
+  }
 }
 
 inline array matrix_norm(
@@ -71,85 +77,77 @@ inline array matrix_norm(
     const std::vector<int>& axis,
     bool keepdims,
     StreamOrDevice s) {
-  if (ord == "f" || ord == "fro")
+  if (ord == "f" || ord == "fro") {
     return sqrt(sum(abs(a, s) * abs(a, s), axis, keepdims, s));
-  else if (ord == "inf")
+  } else if (ord == "nuc") {
-    return matrix_norm(a, 1.0, {axis[1], axis[0]}, keepdims, s);
+    throw std::runtime_error(
-  else if (ord == "-inf")
+        "[linalg::norm] Nuclear norm not yet implemented.");
-    return matrix_norm(a, -1.0, {axis[1], axis[0]}, keepdims, s);
+  } else {
-  if (ord == "nuc")
+    std::ostringstream msg;
-    throw std::logic_error("Nuclear norm is not implemented.");
+    msg << "[linalg::norm] Invalid ord value '" << ord << "' for matrix norm";
-  std::ostringstream error_stream;
+    throw std::invalid_argument(msg.str());
-  error_stream << "Invalid ord value " << ord << " for matrix norm";
+  }
-  throw std::invalid_argument(error_stream.str());
 }
 
 array norm(
     const array& a,
-    const std::vector<int>& axis,
+    const std::optional<std::vector<int>>& axis /* = std::nullopt */,
-    bool keepdims,
+    bool keepdims /* = false */,
-    StreamOrDevice s) {
+    StreamOrDevice s /* = {} */) {
-  auto num_axes = axis.size();
+  if (!axis) {
+    return norm(flatten(a, s), std::vector<int>{0}, keepdims, s);
+  }
 
-  if (num_axes == 0 || num_axes == 1 || num_axes == 2)
+  if (axis.value().size() > 2) {
-    return sqrt(
+    throw std::invalid_argument(
-        sum(abs(a, s) * abs(a, s),
+        "[linalg::norm] Received too many axes for norm");
-            num_axes ? axis : get_reduce_axes({}, a.ndim()),
+  }
-            keepdims,
+  return sqrt(sum(square(a, s), axis.value(), keepdims, s), s);
-            s),
-        s);
-
-  std::ostringstream error_stream;
-  error_stream << "Invalid axis values " << axis;
-  throw std::invalid_argument(error_stream.str());
 }
 
 array norm(
     const array& a,
     const double ord,
-    const std::vector<int>& axis,
+    const std::optional<std::vector<int>>& axis /* = std::nullopt */,
-    bool keepdims,
+    bool keepdims /* = false */,
-    StreamOrDevice s) {
+    StreamOrDevice s /* = {} */) {
-  std::vector<int> ax = axis;
+  std::vector<int> ax;
-
+  if (!axis) {
-  if (axis.empty())
+    ax.resize(a.ndim());
-    ax = get_reduce_axes({}, a.ndim());
+    std::iota(ax.begin(), ax.end(), 0);
-  else
+  } else {
-    ax = normalize_axes(ax, a.ndim());
+    ax = axis.value();
-
+  }
-  auto num_axes = ax.size();
+  if (ax.size() == 1) {
-  if (num_axes == 1)
     return vector_norm(a, ord, ax, keepdims, s);
-  else if (num_axes == 2)
+  } else if (ax.size() == 2) {
     return matrix_norm(a, ord, ax, keepdims, s);
-
+  } else {
-  std::ostringstream error_stream;
+    throw std::invalid_argument(
-  error_stream << "Invalid axis values " << ax;
+        "[linalg::norm] Received too many axes for norm");
-  throw std::invalid_argument(error_stream.str());
+  }
 }
 
 array norm(
     const array& a,
     const std::string& ord,
-    const std::vector<int>& axis,
+    const std::optional<std::vector<int>>& axis /* = std::nullopt */,
-    bool keepdims,
+    bool keepdims /* = false */,
-    StreamOrDevice s) {
+    StreamOrDevice s /* = {} */) {
-  std::vector<int> ax = axis;
+  std::vector<int> ax;
-
+  if (!axis) {
-  if (axis.empty())
+    ax.resize(a.ndim());
-    ax = get_reduce_axes({}, a.ndim());
+    std::iota(ax.begin(), ax.end(), 0);
-  else
+  } else {
-    ax = normalize_axes(ax, a.ndim());
+    ax = axis.value();
-
+  }
-  auto num_axes = ax.size();
+  if (ax.size() != 2) {
-  if (num_axes == 1)
+    std::ostringstream msg;
-    return vector_norm(a, ord, ax, keepdims, s);
+    msg << "[linalg::norm] Norm '" << ord << "' only supported for matrices,"
-  else if (num_axes == 2)
+        << " but received " << ax.size() << " axis/axes.";
-    return matrix_norm(a, ord, ax, keepdims, s);
+    throw std::invalid_argument(msg.str());
-
+  }
-  std::ostringstream error_stream;
+  return matrix_norm(a, ord, ax, keepdims, s);
-  error_stream << "Invalid axis values " << ax;
-  throw std::invalid_argument(error_stream.str());
 }
 
 } // namespace mlx::core::linalg

mlx/linalg.h

@@ -2,27 +2,61 @@
 
 #pragma once
 
+#include <optional>
+
 #include "array.h"
 #include "device.h"
 #include "ops.h"
 #include "stream.h"
 
 namespace mlx::core::linalg {
+
+/*
+ * Compute vector or matrix norms.
+ *
+ * - If axis and ord are both unspecified, computes the 2-norm of flatten(x).
+ * - If axis is not provided but ord is, then x must be either 1D or 2D.
+ * - If axis is provided, but ord is not, then the 2-norm is computed along the
+ *   given axes. At most 2 axes can be specified.
+ * - If both axis and ord are provided, then the corresponding matrix of vector
+ *   norm is computed. At most 2 axes can be specified.
+ */
 array norm(
     const array& a,
     const double ord,
-    const std::vector<int>& axis = {},
+    const std::optional<std::vector<int>>& axis = std::nullopt,
     bool keepdims = false,
     StreamOrDevice s = {});
+inline array norm(
+    const array& a,
+    const double ord,
+    int axis,
+    bool keepdims = false,
+    StreamOrDevice s = {}) {
+  return norm(a, ord, std::vector<int>{axis}, keepdims, s);
+}
 array norm(
     const array& a,
     const std::string& ord,
-    const std::vector<int>& axis = {},
+    const std::optional<std::vector<int>>& axis = std::nullopt,
     bool keepdims = false,
     StreamOrDevice s = {});
+inline array norm(
+    const array& a,
+    const std::string& ord,
+    int axis,
+    bool keepdims = false,
+    StreamOrDevice s = {}) {
+  return norm(a, ord, std::vector<int>{axis}, keepdims, s);
+}
 array norm(
     const array& a,
-    const std::vector<int>& axis = {},
+    const std::optional<std::vector<int>>& axis = std::nullopt,
     bool keepdims = false,
     StreamOrDevice s = {});
+inline array
+norm(const array& a, int axis, bool keepdims = false, StreamOrDevice s = {}) {
+  return norm(a, std::vector<int>{axis}, keepdims, s);
+}
+
 } // namespace mlx::core::linalg

mlx/utils.cpp

@@ -1,6 +1,5 @@
 // Copyright © 2023 Apple Inc.
 
-#include <numeric>
 #include <sstream>
 #include <vector>
 
@@ -74,12 +73,6 @@ int normalize_axis(int axis, int ndim) {
   }
   return axis;
 }
-std::vector<int> normalize_axes(const std::vector<int>& axes, int ndim) {
-  std::vector<int> canonical;
-  for (int ax : axes)
-    canonical.push_back(normalize_axis(ax, ndim));
-  return canonical;
-}
 
 std::ostream& operator<<(std::ostream& os, const Device& d) {
   os << "Device(";

mlx/utils.h

@@ -2,7 +2,6 @@
 
 #pragma once
 
-#include <numeric>
 #include "array.h"
 #include "device.h"
 #include "dtype.h"
@@ -25,7 +24,6 @@ bool is_same_shape(const std::vector<array>& arrays);
  * https://numpy.org/devdocs/reference/generated/numpy.lib.array_utils.normalize_axis_index.html
  */
 int normalize_axis(int axis, int ndim);
-std::vector<int> normalize_axes(const std::vector<int>& axes, int ndim);
 
 std::ostream& operator<<(std::ostream& os, const Device& d);
 std::ostream& operator<<(std::ostream& os, const Stream& s);
@@ -43,18 +41,4 @@ inline std::ostream& operator<<(std::ostream& os, const float16_t& v) {
 inline std::ostream& operator<<(std::ostream& os, const bfloat16_t& v) {
   return os << static_cast<float>(v);
 }
-
-using IntOrVec = std::variant<std::monostate, int, std::vector<int>>;
-inline std::vector<int> get_reduce_axes(const IntOrVec& v, int dims) {
-  std::vector<int> axes;
-  if (std::holds_alternative<std::monostate>(v)) {
-    axes.resize(dims);
-    std::iota(axes.begin(), axes.end(), 0);
-  } else if (auto pv = std::get_if<int>(&v); pv) {
-    axes.push_back(*pv);
-  } else {
-    axes = std::get<std::vector<int>>(v);
-  }
-  return axes;
-}
 } // namespace mlx::core

python/src/fft.cpp

@@ -7,7 +7,6 @@
 
 #include "mlx/fft.h"
 #include "mlx/ops.h"
-#include "mlx/utils.h"
 
 namespace py = pybind11;
 using namespace py::literals;

python/src/linalg.cpp

@@ -26,193 +26,164 @@ using namespace mlx::core;
 using namespace mlx::core::linalg;
 
 void init_linalg(py::module_& parent_module) {
+  py::options options;
+  options.disable_function_signatures();
+
   auto m =
       parent_module.def_submodule("linalg", "mlx.core.linalg: Linear Algebra.");
 
   m.def(
       "norm",
       [](const array& a,
-         const std::variant<std::monostate, int, double, std::string>& ord,
+         const std::variant<std::monostate, int, double, std::string>& ord_,
-         const std::variant<std::monostate, int, std::vector<int>>& axis,
+         const std::variant<std::monostate, int, std::vector<int>>& axis_,
          const bool keepdims,
          const StreamOrDevice stream) {
-        return std::visit(
+        std::optional<std::vector<int>> axis = std::nullopt;
-            overloaded{
+        if (auto pv = std::get_if<int>(&axis_); pv) {
-                [&](const double p) {
+          axis = std::vector<int>{*pv};
-                  if (std::isinf((float)p) || std::isinf(p)) {
+        } else if (auto pv = std::get_if<std::vector<int>>(&axis_); pv) {
-                    if (p > 0) {
+          axis = *pv;
-                      return norm(
+        }
-                          a,
+
-                          "inf",
+        if (std::holds_alternative<std::monostate>(ord_)) {
-                          std::holds_alternative<std::monostate>(axis)
+          return norm(a, axis, keepdims, stream);
-                              ? std::vector<int>()
+        } else {
-                              : get_reduce_axes(axis, a.ndim()),
+          if (auto pv = std::get_if<std::string>(&ord_); pv) {
-                          keepdims,
+            return norm(a, *pv, axis, keepdims, stream);
-                          stream);
+          }
-                    }
+          double ord;
-                    return norm(
+          if (auto pv = std::get_if<int>(&ord_); pv) {
-                        a,
+            ord = *pv;
-                        "-inf",
+          } else {
-                        get_reduce_axes(axis, a.ndim()),
+            ord = std::get<double>(ord_);
-                        keepdims,
+          }
-                        stream);
+          return norm(a, ord, axis, keepdims, stream);
-                  }
+        }
-                  return norm(
-                      a,
-                      p,
-                      std::holds_alternative<std::monostate>(axis)
-                          ? std::vector<int>()
-                          : get_reduce_axes(axis, a.ndim()),
-                      keepdims,
-                      stream);
-                },
-                [&](const std::string& p) {
-                  return norm(
-                      a,
-                      p,
-                      std::holds_alternative<std::monostate>(axis)
-                          ? std::vector<int>()
-                          : get_reduce_axes(axis, a.ndim()),
-                      keepdims,
-                      stream);
-                },
-                [&](const std::monostate _) {
-                  return norm(
-                      a,
-                      std::holds_alternative<std::monostate>(axis)
-                          ? std::vector<int>()
-                          : get_reduce_axes(axis, a.ndim()),
-                      keepdims,
-                      stream);
-                }},
-            ord);
       },
       "a"_a,
+      py::pos_only(),
       "ord"_a = none,
       "axis"_a = none,
       "keepdims"_a = false,
+      py::kw_only(),
       "stream"_a = none,
       R"pbdoc(
-    Matrix or vector norm.
+        norm(a: array, /, ord: Union[None, scalar, str] = None, axis: Union[None, int, List[int]] = None, keepdims: bool = False, *, stream: Union[None, Stream, Device] = None) -> array
 
-    This function is able to return matrix or vector norms,
+        Matrix or vector norm.
-    depending on the value of the ``ord`` parameter.
 
-    Parameters
+        This function computes vector or  matrix norms depending on the value of
-    ----------
+        the ``ord`` and ``axis`` parameters.
-    a : array_like
-        Input array.  If `axis` is None, `a` must be 1-D or 2-D, unless `ord`
-        is None. If both `axis` and `ord` are None, the 2-norm of ``a.flatten`` will be returned.
-    ord : {non-zero int, inf, -inf, 'fro', 'nuc'}, optional
-        Order of the norm (see table under ``Notes``). inf means float(`inf`) object. The default is None.
-    axis : {None, int, 2-tuple of ints}, optional.
-        If `axis` is an integer, it specifies the axis of `a` along which to
-        compute the vector norms.  If `axis` is a 2-tuple, it specifies the
-        axes that hold 2-D matrices, and the matrix norms of these matrices
-        are computed.  If `axis` is None then either a vector norm (when `a`
-        is 1-D) or a matrix norm (when `a` is 2-D) is returned. The default
-        is None.
-    keepdims : bool, optional
-        If this is set to True, the axes which are normed over are left in the
-        result as dimensions with size one.  With this option the result will
-        broadcast correctly against the original `a`.
 
-    Returns
+        Args:
-    -------
+          a (array): Input array.  If ``axis`` is ``None``, ``a`` must be 1-D or 2-D,
-    n : array
+            unless ``ord`` is ``None``. If both ``axis`` and ``ord`` are ``None``, the
-        Norm of the matrix or vector(s).
+            2-norm of ``a.flatten`` will be returned.
+          ord (scalar or str, optional): Order of the norm (see table under ``Notes``).
+            If ``None``, the 2-norm will be computed along the given ``axis``.
+            Default: ``None``.
+          axis (int or list(int), optional): If ``axis`` is an integer, it specifies the
+            axis of ``a`` along which to compute the vector norms.  If ``axis`` is a
+            2-tuple, it specifies the axes that hold 2-D matrices, and the matrix
+            norms of these matrices are computed. If `axis` is ``None`` then
+            either a vector norm (when ``a`` is 1-D) or a matrix norm (when ``a`` is
+            2-D) is returned. Default: ``None``.
+          keepdims (bool, optional): If ``True``, the axes which are normed over are
+            left in the result as dimensions with size one. Default ``False``.
 
-    Notes
+        Returns:
-    -----
+          array: The output containing the norm(s).
-    For values of ``ord < 1``, the result is, strictly speaking, not a
-    mathematical 'norm', but it may still be useful for various numerical
-    purposes.
 
-    The following norms can be calculated:
+        Notes:
+          For values of ``ord < 1``, the result is, strictly speaking, not a
+          mathematical norm, but it may still be useful for various numerical
+          purposes.
 
-    =====  ============================  ==========================
+          The following norms can be calculated:
-    ord    norm for matrices             norm for vectors
-    =====  ============================  ==========================
-    None   Frobenius norm                2-norm
-    'fro'  Frobenius norm                --
-    inf    max(sum(abs(x), axis=1))      max(abs(x))
-    -inf   min(sum(abs(x), axis=1))      min(abs(x))
-    0      --                            sum(x != 0)
-    1      max(sum(abs(x), axis=0))      as below
-    -1     min(sum(abs(x), axis=0))      as below
-    2      2-norm (largest sing. value)  as below
-    -2     smallest singular value       as below
-    other  --                            sum(abs(x)**ord)**(1./ord)
-    =====  ============================  ==========================
 
-    Nuclear norm and norms based on singular values are not yet implemented.
+          =====  ============================  ==========================
+          ord    norm for matrices             norm for vectors
+          =====  ============================  ==========================
+          None   Frobenius norm                2-norm
+          'fro'  Frobenius norm                --
+          inf    max(sum(abs(x), axis=1))      max(abs(x))
+          -inf   min(sum(abs(x), axis=1))      min(abs(x))
+          0      --                            sum(x != 0)
+          1      max(sum(abs(x), axis=0))      as below
+          -1     min(sum(abs(x), axis=0))      as below
+          2      2-norm (largest sing. value)  as below
+          -2     smallest singular value       as below
+          other  --                            sum(abs(x)**ord)**(1./ord)
+          =====  ============================  ==========================
 
-    The Frobenius norm is given by [1]_:
+          .. warning::
+            Nuclear norm and norms based on singular values are not yet implemented.
 
-        :math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
+          The Frobenius norm is given by [1]_:
 
-    The nuclear norm is the sum of the singular values.
+              :math:`||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
 
-    Both the Frobenius and nuclear norm orders are only defined for
+          The nuclear norm is the sum of the singular values.
-    matrices and raise a ValueError when ``a.ndim != 2``.
 
-    References
+          Both the Frobenius and nuclear norm orders are only defined for
-    ----------
+          matrices and raise a ``ValueError`` when ``a.ndim != 2``.
-    .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
-           Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
 
-    Examples
+        References:
-    --------
+          .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
-    >>> import mlx.core as mx
+                 Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
-    >>> from mlx.core import linalg as LA
+
-    >>> a = mx.arange(9) - 4
+        Examples:
-    >>> a
+          >>> import mlx.core as mx
-    array([-4, -3, -2, ..., 2, 3, 4], dtype=int32)
+          >>> from mlx.core import linalg as la
-    >>> b = a.reshape((3,3))
+          >>> a = mx.arange(9) - 4
-    >>> b
+          >>> a
-    array([[-4, -3, -2],
+          array([-4, -3, -2, ..., 2, 3, 4], dtype=int32)
-           [-1,  0,  1],
+          >>> b = a.reshape((3,3))
-           [ 2,  3,  4]], dtype=int32)
+          >>> b
-    >>> LA.norm(a)
+          array([[-4, -3, -2],
-    array(7.74597, dtype=float32)
+                 [-1,  0,  1],
-    >>> LA.norm(b)
+                 [ 2,  3,  4]], dtype=int32)
-    array(7.74597, dtype=float32)
+          >>> la.norm(a)
-    >>> LA.norm(b, 'fro')
+          array(7.74597, dtype=float32)
-    array(7.74597, dtype=float32)
+          >>> la.norm(b)
-    >>> LA.norm(a, float("inf"))
+          array(7.74597, dtype=float32)
-    array(4, dtype=int32)
+          >>> la.norm(b, 'fro')
-    >>> LA.norm(b, float("inf"))
+          array(7.74597, dtype=float32)
-    array(9, dtype=int32)
+          >>> la.norm(a, float("inf"))
-    >>> LA.norm(a, -float("inf"))
+          array(4, dtype=float32)
-    array(0, dtype=int32)
+          >>> la.norm(b, float("inf"))
-    >>> LA.norm(b, -float("inf"))
+          array(9, dtype=float32)
-    array(2, dtype=int32)
+          >>> la.norm(a, -float("inf"))
-    >>> LA.norm(a, 1)
+          array(0, dtype=float32)
-    array(20, dtype=int32)
+          >>> la.norm(b, -float("inf"))
-    >>> LA.norm(b, 1)
+          array(2, dtype=float32)
-    array(7, dtype=int32)
+          >>> la.norm(a, 1)
-    >>> LA.norm(a, -1)
+          array(20, dtype=float32)
-    array(0, dtype=float32)
+          >>> la.norm(b, 1)
-    >>> LA.norm(b, -1)
+          array(7, dtype=float32)
-    array(6, dtype=int32)
+          >>> la.norm(a, -1)
-    >>> LA.norm(a, 2)
+          array(0, dtype=float32)
-    array(7.74597, dtype=float32)
+          >>> la.norm(b, -1)
-    >>> LA.norm(a, 3)
+          array(6, dtype=float32)
-    array(5.84804, dtype=float32)
+          >>> la.norm(a, 2)
-    >>> LA.norm(a, -3)
+          array(7.74597, dtype=float32)
-    array(0, dtype=float32)
+          >>> la.norm(a, 3)
-    >>> c = mx.array([[ 1, 2, 3],
+          array(5.84804, dtype=float32)
-    ...               [-1, 1, 4]])
+          >>> la.norm(a, -3)
-    >>> LA.norm(c, axis=0)
+          array(0, dtype=float32)
-    array([1.41421, 2.23607, 5], dtype=float32)
+          >>> c = mx.array([[ 1, 2, 3],
-    >>> LA.norm(c, axis=1)
+          ...               [-1, 1, 4]])
-    array([3.74166, 4.24264], dtype=float32)
+          >>> la.norm(c, axis=0)
-    >>> LA.norm(c, ord=1, axis=1)
+          array([1.41421, 2.23607, 5], dtype=float32)
-    array([6, 6], dtype=int32)
+          >>> la.norm(c, axis=1)
-    >>> m = mx.arange(8).reshape(2,2,2)
+          array([3.74166, 4.24264], dtype=float32)
-    >>> LA.norm(m, axis=(1,2))
+          >>> la.norm(c, ord=1, axis=1)
-    array([3.74166, 11.225], dtype=float32)
+          array([6, 6], dtype=float32)
-    >>> LA.norm(m[0, :, :]), LA.norm(m[1, :, :])
+          >>> m = mx.arange(8).reshape(2,2,2)
-    (array(3.74166, dtype=float32), array(11.225, dtype=float32))
+          >>> la.norm(m, axis=(1,2))
-    )pbdoc");
+          array([3.74166, 11.225], dtype=float32)
+          >>> la.norm(m[0, :, :]), LA.norm(m[1, :, :])
+          (array(3.74166, dtype=float32), array(11.225, dtype=float32))
+      )pbdoc");
 }

python/src/utils.h

@@ -1,6 +1,7 @@
 // Copyright © 2023 Apple Inc.
 
 #pragma once
+#include <numeric>
 #include <variant>
 
 #include <pybind11/complex.h>
@@ -13,10 +14,24 @@ namespace py = pybind11;
 
 using namespace mlx::core;
 
+using IntOrVec = std::variant<std::monostate, int, std::vector<int>>;
 using ScalarOrArray = std::
     variant<py::bool_, py::int_, py::float_, std::complex<float>, py::object>;
 static constexpr std::monostate none{};
 
+inline std::vector<int> get_reduce_axes(const IntOrVec& v, int dims) {
+  std::vector<int> axes;
+  if (std::holds_alternative<std::monostate>(v)) {
+    axes.resize(dims);
+    std::iota(axes.begin(), axes.end(), 0);
+  } else if (auto pv = std::get_if<int>(&v); pv) {
+    axes.push_back(*pv);
+  } else {
+    axes = std::get<std::vector<int>>(v);
+  }
+  return axes;
+}
+
 inline array to_array_with_accessor(py::object obj) {
   if (py::hasattr(obj, "__mlx_array__")) {
     return obj.attr("__mlx_array__")().cast<array>();

tests/linalg_tests.cpp

@@ -3,170 +3,155 @@
 #include "doctest/doctest.h"
 
 #include <cmath>
-#include <iostream>
+
-#include "mlx/linalg.h"
 #include "mlx/mlx.h"
 
 using namespace mlx::core;
 using namespace mlx::core::linalg;
 
 TEST_CASE("[mlx.core.linalg.norm] no ord") {
-  array arr_one_d({1, 2, 3});
+  // Zero dimensions
-  array arr_two_d = reshape(arange(9), {3, 3});
+  array x(2.0);
-  array arr_three_d = reshape(arange(18), {2, 3, 3});
+  CHECK_EQ(norm(x).item<float>(), 2.0f);
+  CHECK_THROWS(norm(x, 0));
 
-  CHECK(array_equal(norm(arr_one_d), array(sqrt(1 + 4 + 9))).item<bool>());
+  x = array({1, 2, 3});
-  CHECK(array_equal(norm(arr_one_d, {0}, false), array(sqrt(1 + 4 + 9)))
+  float expected = std::sqrt(1 + 4 + 9);
-            .item<bool>());
+  CHECK_EQ(norm(x).item<float>(), doctest::Approx(expected));
+  CHECK_EQ(norm(x, 0, false).item<float>(), doctest::Approx(expected));
+  CHECK_EQ(norm(x, -1, false).item<float>(), doctest::Approx(expected));
+  CHECK_EQ(norm(x, -1, true).ndim(), 1);
+  CHECK_THROWS(norm(x, 1));
+
+  x = reshape(arange(9), {3, 3});
+  expected =
+      std::sqrt(0 + 1 + 2 * 2 + 3 * 3 + 4 * 4 + 5 * 5 + 6 * 6 + 7 * 7 + 8 * 8);
+
+  CHECK_EQ(norm(x).item<float>(), doctest::Approx(expected));
+  CHECK_EQ(
+      norm(x, std::vector<int>{0, 1}).item<float>(), doctest::Approx(expected));
   CHECK(array_equal(
-            norm(arr_two_d, {}, false),
+            norm(x, 0, false),
-            array(sqrt(
-                0 + 1 + 2 * 2 + 3 * 3 + 4 * 4 + 5 * 5 + 6 * 6 + 7 * 7 + 8 * 8)))
-            .item<bool>());
-  CHECK(array_equal(
-            norm(arr_two_d, {0}, false),
             array(
-                {sqrt(0 + 3 * 3 + 6 * 6),
+                {std::sqrt(0 + 3 * 3 + 6 * 6),
-                 sqrt(1 + 4 * 4 + 7 * 7),
+                 std::sqrt(1 + 4 * 4 + 7 * 7),
-                 sqrt(2 * 2 + 5 * 5 + 8 * 8)}))
+                 std::sqrt(2 * 2 + 5 * 5 + 8 * 8)}))
             .item<bool>());
-  CHECK(array_equal(
+  CHECK(allclose(
-            norm(arr_two_d, {1}, false),
+            norm(x, 1, false),
             array(
-                {sqrt(0 + 1 + 2 * 2),
+                {std::sqrt(0 + 1 + 2 * 2),
-                 sqrt(3 * 3 + 4 * 4 + 5 * 5),
+                 std::sqrt(3 * 3 + 4 * 4 + 5 * 5),
-                 sqrt(6 * 6 + 7 * 7 + 8 * 8)}))
+                 std::sqrt(6 * 6 + 7 * 7 + 8 * 8)}))
             .item<bool>());
-  CHECK(array_equal(
+
-            norm(arr_two_d, {0, 1}, false),
+  x = reshape(arange(18), {2, 3, 3});
-            array(sqrt(
+  CHECK(allclose(
-                0 + 1 + 2 * 2 + 3 * 3 + 4 * 4 + 5 * 5 + 6 * 6 + 7 * 7 + 8 * 8)))
+            norm(x, 2, false),
-            .item<bool>());
-  CHECK(array_equal(
-            norm(arr_three_d, {2}, false),
             array(
                 {
-                    sqrt(0 + 1 + 2 * 2),
+                    std::sqrt(0 + 1 + 2 * 2),
-                    sqrt(3 * 3 + 4 * 4 + 5 * 5),
+                    std::sqrt(3 * 3 + 4 * 4 + 5 * 5),
-                    sqrt(6 * 6 + 7 * 7 + 8 * 8),
+                    std::sqrt(6 * 6 + 7 * 7 + 8 * 8),
-                    sqrt(9 * 9 + 10 * 10 + 11 * 11),
+                    std::sqrt(9 * 9 + 10 * 10 + 11 * 11),
-                    sqrt(12 * 12 + 13 * 13 + 14 * 14),
+                    std::sqrt(12 * 12 + 13 * 13 + 14 * 14),
-                    sqrt(15 * 15 + 16 * 16 + 17 * 17),
+                    std::sqrt(15 * 15 + 16 * 16 + 17 * 17),
                 },
                 {2, 3}))
             .item<bool>());
-  CHECK(array_equal(
+  CHECK(allclose(
-            norm(arr_three_d, {1}, false),
+            norm(x, std::vector<int>{1, 2}, false),
             array(
-                {
+                {std::sqrt(
-                    sqrt(0 + 3 * 3 + 6 * 6),
-                    sqrt(1 + 4 * 4 + 7 * 7),
-                    sqrt(2 * 2 + 5 * 5 + 8 * 8),
-                    sqrt(9 * 9 + 12 * 12 + 15 * 15),
-                    sqrt(10 * 10 + 13 * 13 + 16 * 16),
-                    sqrt(11 * 11 + 14 * 14 + 17 * 17),
-                },
-                {2, 3}))
-            .item<bool>());
-  CHECK(array_equal(
-            norm(arr_three_d, {0}, false),
-            array(
-                {
-                    sqrt(0 + 9 * 9),
-                    sqrt(1 + 10 * 10),
-                    sqrt(2 * 2 + 11 * 11),
-                    sqrt(3 * 3 + 12 * 12),
-                    sqrt(4 * 4 + 13 * 13),
-                    sqrt(5 * 5 + 14 * 14),
-                    sqrt(6 * 6 + 15 * 15),
-                    sqrt(7 * 7 + 16 * 16),
-                    sqrt(8 * 8 + 17 * 17),
-                },
-                {3, 3}))
-            .item<bool>());
-  CHECK(array_equal(
-            norm(arr_three_d, {1, 2}, false),
-            array(
-                {sqrt(
                      0 + 1 + 2 * 2 + 3 * 3 + 4 * 4 + 5 * 5 + 6 * 6 + 7 * 7 +
                      8 * 8),
-                 sqrt(
+                 std::sqrt(
                      9 * 9 + 10 * 10 + 11 * 11 + 12 * 12 + 13 * 13 + 14 * 14 +
                      15 * 15 + 16 * 16 + 17 * 17)},
                 {2}))
             .item<bool>());
+  CHECK_THROWS(norm(x, std::vector<int>{0, 1, 2}));
 }
 
 TEST_CASE("[mlx.core.linalg.norm] double ord") {
-  array arr_one_d({1, 2, 3});
+  CHECK_THROWS(norm(array(0), 2.0));
-  array arr_two_d = reshape(arange(9), {3, 3});
-  array arr_three_d = reshape(arange(18), {2, 3, 3});
 
-  CHECK(array_equal(norm(arr_one_d, 2.0), array(sqrt(1 + 4 + 9))).item<bool>());
+  array x({1, 2, 3});
-  CHECK(array_equal(norm(arr_one_d, 1.0), array(1 + 2 + 3)).item<bool>());
-  CHECK(array_equal(norm(arr_one_d, 0.0), array(3)).item<bool>());
 
-  CHECK(array_equal(norm(arr_one_d, 2.0, {0}, false), array(sqrt(1 + 4 + 9)))
+  float expected = std::sqrt(1 + 4 + 9);
-            .item<bool>());
+  CHECK_EQ(norm(x, 2.0).item<float>(), doctest::Approx(expected));
-  CHECK(array_equal(
+  CHECK_EQ(norm(x, 2.0, 0).item<float>(), doctest::Approx(expected));
-            norm(arr_two_d, 2.0, {0}, false),
+  CHECK_THROWS(norm(x, 2.0, 1));
+
+  expected = 1 + 2 + 3;
+  CHECK_EQ(norm(x, 1.0).item<float>(), doctest::Approx(expected));
+
+  expected = 3;
+  CHECK_EQ(norm(x, 0.0).item<float>(), doctest::Approx(expected));
+
+  expected = 3;
+  CHECK_EQ(
+      norm(x, std::numeric_limits<double>::infinity()).item<float>(),
+      doctest::Approx(expected));
+
+  expected = 1;
+  CHECK_EQ(
+      norm(x, -std::numeric_limits<double>::infinity()).item<float>(),
+      doctest::Approx(expected));
+
+  x = reshape(arange(9), {3, 3});
+
+  CHECK(allclose(
+            norm(x, 2.0, 0, false),
             array(
-                {sqrt(0 + 3 * 3 + 6 * 6),
+                {std::sqrt(0 + 3 * 3 + 6 * 6),
-                 sqrt(1 + 4 * 4 + 7 * 7),
+                 std::sqrt(1 + 4 * 4 + 7 * 7),
-                 sqrt(2 * 2 + 5 * 5 + 8 * 8)}))
+                 std::sqrt(2 * 2 + 5 * 5 + 8 * 8)}))
             .item<bool>());
-  CHECK(array_equal(
+  CHECK(allclose(
-            norm(arr_two_d, 2.0, {1}, false),
+            norm(x, 2.0, 1, false),
             array(
                 {sqrt(0 + 1 + 2 * 2),
                  sqrt(3 * 3 + 4 * 4 + 5 * 5),
                  sqrt(6 * 6 + 7 * 7 + 8 * 8)}))
             .item<bool>());
-  CHECK(array_equal(
-            norm(arr_three_d, 2.0, {2}, false),
-            array(
-                {
-                    sqrt(0 + 1 + 2 * 2),
-                    sqrt(3 * 3 + 4 * 4 + 5 * 5),
-                    sqrt(6 * 6 + 7 * 7 + 8 * 8),
-                    sqrt(9 * 9 + 10 * 10 + 11 * 11),
-                    sqrt(12 * 12 + 13 * 13 + 14 * 14),
-                    sqrt(15 * 15 + 16 * 16 + 17 * 17),
-                },
-                {2, 3}))
-            .item<bool>());
-  CHECK(array_equal(
-            norm(arr_three_d, 2.0, {1}, false),
-            array(
-                {
-                    sqrt(0 + 3 * 3 + 6 * 6),
-                    sqrt(1 + 4 * 4 + 7 * 7),
-                    sqrt(2 * 2 + 5 * 5 + 8 * 8),
-                    sqrt(9 * 9 + 12 * 12 + 15 * 15),
-                    sqrt(10 * 10 + 13 * 13 + 16 * 16),
-                    sqrt(11 * 11 + 14 * 14 + 17 * 17),
-                },
-                {2, 3}))
-            .item<bool>());
-  CHECK(array_equal(
-            norm(arr_three_d, 2.0, {0}, false),
-            array(
-                {
-                    sqrt(0 + 9 * 9),
-                    sqrt(1 + 10 * 10),
-                    sqrt(2 * 2 + 11 * 11),
-                    sqrt(3 * 3 + 12 * 12),
-                    sqrt(4 * 4 + 13 * 13),
-                    sqrt(5 * 5 + 14 * 14),
-                    sqrt(6 * 6 + 15 * 15),
-                    sqrt(7 * 7 + 16 * 16),
-                    sqrt(8 * 8 + 17 * 17),
-                },
-                {3, 3}))
-            .item<bool>());
 
+  CHECK_EQ(
+      norm(x, 1.0, std::vector<int>{0, 1}).item<float>(),
+      doctest::Approx(15.0));
+  CHECK_EQ(
+      norm(x, 1.0, std::vector<int>{1, 0}).item<float>(),
+      doctest::Approx(21.0));
+  CHECK_EQ(
+      norm(x, -1.0, std::vector<int>{0, 1}).item<float>(),
+      doctest::Approx(9.0));
+  CHECK_EQ(
+      norm(x, -1.0, std::vector<int>{1, 0}).item<float>(),
+      doctest::Approx(3.0));
+  CHECK_EQ(
+      norm(x, 1.0, std::vector<int>{0, 1}, true).shape(),
+      std::vector<int>{1, 1});
+  CHECK_EQ(
+      norm(x, 1.0, std::vector<int>{1, 0}, true).shape(),
+      std::vector<int>{1, 1});
+  CHECK_EQ(
+      norm(x, -1.0, std::vector<int>{0, 1}, true).shape(),
+      std::vector<int>{1, 1});
+  CHECK_EQ(
+      norm(x, -1.0, std::vector<int>{1, 0}, true).shape(),
+      std::vector<int>{1, 1});
+
+  CHECK_EQ(
+      norm(x, -1.0, std::vector<int>{-2, -1}, false).item<float>(),
+      doctest::Approx(9.0));
+  CHECK_EQ(
+      norm(x, 1.0, std::vector<int>{-2, -1}, false).item<float>(),
+      doctest::Approx(15.0));
+
+  x = reshape(arange(18), {2, 3, 3});
+  CHECK_THROWS(norm(x, 2.0, std::vector{0, 1, 2}));
   CHECK(allclose(
-            norm(arr_three_d, 3.0, {0}),
+            norm(x, 3.0, 0),
             array(
                 {9.,
                  10.00333222,
@@ -179,15 +164,8 @@ TEST_CASE("[mlx.core.linalg.norm] double ord") {
                  17.57113899},
                 {3, 3}))
             .item<bool>());
-  CHECK(
-      allclose(
-          norm(arr_three_d, 3.0, {1}),
-          array(
-              {6.24025147, 7.41685954, 8.6401226, 18., 19.39257164, 20.7915893},
-              {2, 3}))
-          .item<bool>());
   CHECK(allclose(
-            norm(arr_three_d, 3.0, {2}),
+            norm(x, 3.0, 2),
             array(
                 {2.08008382,
                  6.,
@@ -197,110 +175,76 @@ TEST_CASE("[mlx.core.linalg.norm] double ord") {
                  23.13593104},
                 {2, 3}))
             .item<bool>());
-  CHECK(allclose(
+  CHECK(
-            norm(arr_three_d, 0.0, {0}),
+      allclose(
-            array({1., 2., 2., 2., 2., 2., 2., 2., 2.}, {3, 3}))
+          norm(x, 0.0, 0), array({1., 2., 2., 2., 2., 2., 2., 2., 2.}, {3, 3}))
+          .item<bool>());
+  CHECK(allclose(norm(x, 0.0, 1), array({2., 3., 3., 3., 3., 3.}, {2, 3}))
+            .item<bool>());
+  CHECK(allclose(norm(x, 0.0, 2), array({2., 3., 3., 3., 3., 3.}, {2, 3}))
             .item<bool>());
-  CHECK(
-      allclose(
-          norm(arr_three_d, 0.0, {1}), array({2., 3., 3., 3., 3., 3.}, {2, 3}))
-          .item<bool>());
-  CHECK(
-      allclose(
-          norm(arr_three_d, 0.0, {2}), array({2., 3., 3., 3., 3., 3.}, {2, 3}))
-          .item<bool>());
   CHECK(allclose(
-            norm(arr_three_d, 1.0, {0}),
+            norm(x, 1.0, 0),
             array({9., 11., 13., 15., 17., 19., 21., 23., 25.}, {3, 3}))
             .item<bool>());
-  CHECK(allclose(
+  CHECK(allclose(norm(x, 1.0, 1), array({9., 12., 15., 36., 39., 42.}, {2, 3}))
-            norm(arr_three_d, 1.0, {1}),
-            array({9., 12., 15., 36., 39., 42.}, {2, 3}))
             .item<bool>());
-  CHECK(allclose(
+  CHECK(allclose(norm(x, 1.0, 2), array({3., 12., 21., 30., 39., 48.}, {2, 3}))
-            norm(arr_three_d, 1.0, {2}),
-            array({3., 12., 21., 30., 39., 48.}, {2, 3}))
             .item<bool>());
 
-  CHECK(allclose(norm(arr_two_d, 1.0, {0, 1}), array({15.0})).item<bool>());
+  CHECK(allclose(norm(x, 1.0, std::vector<int>{0, 1}), array({21., 23., 25.}))
-  CHECK(allclose(norm(arr_two_d, 1.0, {1, 0}), array({21.0})).item<bool>());
-  CHECK(allclose(norm(arr_two_d, -1.0, {0, 1}), array({9.0})).item<bool>());
-  CHECK(allclose(norm(arr_two_d, -1.0, {1, 0}), array({3.0})).item<bool>());
-
-  CHECK(allclose(norm(arr_two_d, 1.0, {0, 1}, true), array({15.0}, {1, 1}))
             .item<bool>());
-  CHECK(allclose(norm(arr_two_d, 1.0, {1, 0}, true), array({21.0}, {1, 1}))
+  CHECK(allclose(norm(x, 1.0, std::vector<int>{1, 2}), array({15., 42.}))
             .item<bool>());
-  CHECK(allclose(norm(arr_two_d, -1.0, {0, 1}, true), array({9.0}, {1, 1}))
+  CHECK(allclose(norm(x, -1.0, std::vector<int>{0, 1}), array({9., 11., 13.}))
             .item<bool>());
-  CHECK(allclose(norm(arr_two_d, -1.0, {1, 0}, true), array({3.0}, {1, 1}))
+  CHECK(allclose(norm(x, -1.0, std::vector<int>{1, 2}), array({9., 36.}))
             .item<bool>());
-
+  CHECK(allclose(norm(x, -1.0, std::vector<int>{1, 0}), array({9., 12., 15.}))
-  CHECK(array_equal(norm(arr_two_d, -1.0, {-2, -1}, false), array(9.0))
             .item<bool>());
-  CHECK(array_equal(norm(arr_two_d, 1.0, {-2, -1}, false), array(15.0))
+  CHECK(allclose(norm(x, -1.0, std::vector<int>{2, 1}), array({3, 30}))
             .item<bool>());
-  //
+  CHECK(allclose(norm(x, -1.0, std::vector<int>{1, 2}), array({9, 36}))
-  CHECK(allclose(norm(arr_three_d, 1.0, {0, 1}), array({21., 23., 25.}))
             .item<bool>());
-  CHECK(
-      allclose(norm(arr_three_d, 1.0, {1, 2}), array({15., 42.})).item<bool>());
-  CHECK(allclose(norm(arr_three_d, -1.0, {0, 1}), array({9., 11., 13.}))
-            .item<bool>());
-  CHECK(
-      allclose(norm(arr_three_d, -1.0, {1, 2}), array({9., 36.})).item<bool>());
-  CHECK(allclose(norm(arr_three_d, -1.0, {1, 0}), array({9., 12., 15.}))
-            .item<bool>());
-  CHECK(allclose(norm(arr_three_d, -1.0, {2, 1}), array({3, 30})).item<bool>());
-  CHECK(allclose(norm(arr_three_d, -1.0, {1, 2}), array({9, 36})).item<bool>());
 }
 
 TEST_CASE("[mlx.core.linalg.norm] string ord") {
-  array arr_one_d({1, 2, 3});
+  array x({1, 2, 3});
-  array arr_two_d = reshape(arange(9), {3, 3});
+  CHECK_THROWS(norm(x, "fro"));
-  array arr_three_d = reshape(arange(18), {2, 3, 3});
 
-  CHECK(allclose(norm(arr_one_d, "inf", {}), array({3.0})).item<bool>());
+  x = reshape(arange(9), {3, 3});
-  CHECK(allclose(norm(arr_one_d, "-inf", {}), array({1.0})).item<bool>());
+  CHECK_THROWS(norm(x, "bad ord"));
 
-  CHECK(allclose(norm(arr_two_d, "f", {0, 1}), array({14.2828568570857}))
+  CHECK_EQ(
-            .item<bool>());
+      norm(x, "f", std::vector<int>{0, 1}).item<float>(),
-  CHECK(allclose(norm(arr_two_d, "fro", {0, 1}), array({14.2828568570857}))
+      doctest::Approx(14.2828568570857));
-            .item<bool>());
+  CHECK_EQ(
-  CHECK(allclose(norm(arr_two_d, "inf", {0, 1}), array({21.0})).item<bool>());
+      norm(x, "fro", std::vector<int>{0, 1}).item<float>(),
-  CHECK(allclose(norm(arr_two_d, "-inf", {0, 1}), array({3.0})).item<bool>());
+      doctest::Approx(14.2828568570857));
 
+  x = reshape(arange(18), {2, 3, 3});
   CHECK(allclose(
-            norm(arr_three_d, "fro", {0, 1}),
+            norm(x, "fro", std::vector<int>{0, 1}),
             array({22.24859546, 24.31049156, 26.43860813}))
             .item<bool>());
   CHECK(allclose(
-            norm(arr_three_d, "fro", {1, 2}), array({14.28285686, 39.7617907}))
+            norm(x, "fro", std::vector<int>{1, 2}),
+            array({14.28285686, 39.7617907}))
             .item<bool>());
   CHECK(allclose(
-            norm(arr_three_d, "f", {0, 1}),
+            norm(x, "f", std::vector<int>{0, 1}),
             array({22.24859546, 24.31049156, 26.43860813}))
             .item<bool>());
   CHECK(allclose(
-            norm(arr_three_d, "f", {1, 0}),
+            norm(x, "f", std::vector<int>{1, 0}),
             array({22.24859546, 24.31049156, 26.43860813}))
             .item<bool>());
-  CHECK(
+  CHECK(allclose(
-      allclose(norm(arr_three_d, "f", {1, 2}), array({14.28285686, 39.7617907}))
+            norm(x, "f", std::vector<int>{1, 2}),
-          .item<bool>());
+            array({14.28285686, 39.7617907}))
-  CHECK(
-      allclose(norm(arr_three_d, "f", {2, 1}), array({14.28285686, 39.7617907}))
-          .item<bool>());
-  CHECK(allclose(norm(arr_three_d, "inf", {0, 1}), array({36., 39., 42.}))
             .item<bool>());
-  CHECK(allclose(norm(arr_three_d, "inf", {1, 2}), array({21., 48.}))
+  CHECK(allclose(
-            .item<bool>());
+            norm(x, "f", std::vector<int>{2, 1}),
-  CHECK(allclose(norm(arr_three_d, "-inf", {0, 1}), array({9., 12., 15.}))
+            array({14.28285686, 39.7617907}))
-            .item<bool>());
-  CHECK(allclose(norm(arr_three_d, "-inf", {1, 2}), array({3., 30.}))
-            .item<bool>());
-  CHECK(allclose(norm(arr_three_d, "-inf", {1, 0}), array({9., 11., 13.}))
-            .item<bool>());
-  CHECK(allclose(norm(arr_three_d, "-inf", {2, 1}), array({9., 36.}))
             .item<bool>());
 }