CPU LU factorization and linear solvers (#1451)

* linalg solve backend

* nits

* more nits + fix

* luf primitive and lu, solve, and solve_triangular backends

* changes / nits

---------

Co-authored-by: Awni Hannun <awni@apple.com>

mlx/backend/cpu/CMakeLists.txt

@@ -59,6 +59,7 @@ target_sources(
           ${CMAKE_CURRENT_SOURCE_DIR}/sort.cpp
           ${CMAKE_CURRENT_SOURCE_DIR}/threefry.cpp
           ${CMAKE_CURRENT_SOURCE_DIR}/indexing.cpp
+          ${CMAKE_CURRENT_SOURCE_DIR}/luf.cpp
           ${CMAKE_CURRENT_SOURCE_DIR}/qrf.cpp
           ${CMAKE_CURRENT_SOURCE_DIR}/svd.cpp
           ${CMAKE_CURRENT_SOURCE_DIR}/inverse.cpp

mlx/backend/cpu/luf.cpp

@@ -0,0 +1,88 @@
+// Copyright © 2024 Apple Inc.
+
+#include <cassert>
+
+#include "mlx/allocator.h"
+#include "mlx/backend/cpu/copy.h"
+#include "mlx/backend/cpu/lapack.h"
+#include "mlx/primitives.h"
+
+namespace mlx::core {
+
+void lu_factor_impl(
+    const array& a,
+    array& lu,
+    array& pivots,
+    array& row_indices) {
+  int M = a.shape(-2);
+  int N = a.shape(-1);
+
+  // Copy a into lu and make it col contiguous
+  auto ndim = lu.ndim();
+  auto flags = lu.flags();
+  flags.col_contiguous = ndim == 2;
+  flags.row_contiguous = false;
+  flags.contiguous = true;
+  auto strides = lu.strides();
+  strides[ndim - 1] = M;
+  strides[ndim - 2] = 1;
+  lu.set_data(
+      allocator::malloc_or_wait(lu.nbytes()), lu.nbytes(), strides, flags);
+  copy_inplace(
+      a, lu, a.shape(), a.strides(), strides, 0, 0, CopyType::GeneralGeneral);
+
+  auto a_ptr = lu.data<float>();
+
+  pivots.set_data(allocator::malloc_or_wait(pivots.nbytes()));
+  row_indices.set_data(allocator::malloc_or_wait(row_indices.nbytes()));
+  auto pivots_ptr = pivots.data<uint32_t>();
+  auto row_indices_ptr = row_indices.data<uint32_t>();
+
+  int info;
+  size_t num_matrices = a.size() / (M * N);
+  for (size_t i = 0; i < num_matrices; ++i) {
+    // Compute LU factorization of A
+    MLX_LAPACK_FUNC(sgetrf)
+    (/* m */ &M,
+     /* n */ &N,
+     /* a */ a_ptr,
+     /* lda */ &M,
+     /* ipiv */ reinterpret_cast<int*>(pivots_ptr),
+     /* info */ &info);
+
+    if (info != 0) {
+      std::stringstream ss;
+      ss << "[LUF::eval_cpu] sgetrf_ failed with code " << info
+         << ((info > 0) ? " because matrix is singular"
+                        : " because argument had an illegal value");
+      throw std::runtime_error(ss.str());
+    }
+
+    // Subtract 1 to get 0-based index
+    for (int j = 0; j < pivots.shape(-1); ++j) {
+      pivots_ptr[j]--;
+      row_indices_ptr[j] = j;
+    }
+    for (int j = pivots.shape(-1) - 1; j >= 0; --j) {
+      auto piv = pivots_ptr[j];
+      auto t1 = row_indices_ptr[piv];
+      auto t2 = row_indices_ptr[j];
+      row_indices_ptr[j] = t1;
+      row_indices_ptr[piv] = t2;
+    }
+
+    // Advance pointers to the next matrix
+    a_ptr += M * N;
+    pivots_ptr += pivots.shape(-1);
+    row_indices_ptr += pivots.shape(-1);
+  }
+}
+
+void LUF::eval_cpu(
+    const std::vector<array>& inputs,
+    std::vector<array>& outputs) {
+  assert(inputs.size() == 1);
+  lu_factor_impl(inputs[0], outputs[0], outputs[1], outputs[2]);
+}
+
+} // namespace mlx::core

mlx/backend/metal/primitives.cpp

@@ -554,6 +554,12 @@ void Eigh::eval_gpu(
   throw std::runtime_error("[Eigvalsh::eval_gpu] Metal Eigh NYI.");
 }
 
+void LUF::eval_gpu(
+    const std::vector<array>& inputs,
+    std::vector<array>& outputs) {
+  throw std::runtime_error("[LUF::eval_gpu] Metal LU factorization NYI.");
+}
+
 void View::eval_gpu(const std::vector<array>& inputs, array& out) {
   auto& in = inputs[0];
   auto ibytes = size_of(in.dtype());

mlx/backend/no_cpu/primitives.cpp

@@ -81,6 +81,7 @@ NO_CPU(LogicalNot)
 NO_CPU(LogicalAnd)
 NO_CPU(LogicalOr)
 NO_CPU(LogAddExp)
+NO_CPU_MULTI(LUF)
 NO_CPU(Matmul)
 NO_CPU(Maximum)
 NO_CPU(Minimum)

mlx/backend/no_metal/primitives.cpp

@@ -81,6 +81,7 @@ NO_GPU(LogicalNot)
 NO_GPU(LogicalAnd)
 NO_GPU(LogicalOr)
 NO_GPU(LogAddExp)
+NO_GPU_MULTI(LUF)
 NO_GPU(Matmul)
 NO_GPU(Maximum)
 NO_GPU(Minimum)

mlx/linalg.cpp

@@ -282,7 +282,7 @@ array inv(const array& a, StreamOrDevice s /* = {} */) {
 
 array tri_inv(
     const array& a,
-    bool upper /* = true */,
+    bool upper /* = false */,
     StreamOrDevice s /* = {} */) {
   return inv_impl(a, /*tri=*/true, upper, s);
 }
@@ -519,4 +519,134 @@ std::pair<array, array> eigh(
   return std::make_pair(out[0], out[1]);
 }
 
+void validate_lu(
+    const array& a,
+    const StreamOrDevice& stream,
+    const std::string& fname) {
+  check_cpu_stream(stream, fname);
+  if (a.dtype() != float32) {
+    std::ostringstream msg;
+    msg << fname << " Arrays must type float32. Received array "
+        << "with type " << a.dtype() << ".";
+    throw std::invalid_argument(msg.str());
+  }
+
+  if (a.ndim() < 2) {
+    std::ostringstream msg;
+    msg << fname
+        << " Arrays must have >= 2 dimensions. Received array "
+           "with "
+        << a.ndim() << " dimensions.";
+    throw std::invalid_argument(msg.str());
+  }
+
+  if (a.shape(-1) != a.shape(-2)) {
+    throw std::invalid_argument(fname + " Only defined for square matrices.");
+  }
+}
+
+std::vector<array> lu_helper(const array& a, StreamOrDevice s /* = {} */) {
+  int m = a.shape()[a.shape().size() - 2];
+  int n = a.shape()[a.shape().size() - 1];
+
+  Shape pivots_shape(a.shape().begin(), a.shape().end() - 2);
+  pivots_shape.push_back(std::min(m, n));
+
+  return array::make_arrays(
+      {a.shape(), pivots_shape, pivots_shape},
+      {a.dtype(), uint32, uint32},
+      std::make_shared<LUF>(to_stream(s)),
+      {astype(a, a.dtype(), s)});
+}
+
+std::vector<array> lu(const array& a, StreamOrDevice s /* = {} */) {
+  validate_lu(a, s, "[linalg::lu]");
+
+  auto out = lu_helper(a, s);
+  auto& LU = out[0];
+  auto& row_pivots = out[2];
+
+  int N = a.shape(-1);
+  auto L = add(tril(LU, /* k = */ -1, s), eye(N, s), s);
+  auto U = triu(LU, /* k = */ 0, s);
+  return {row_pivots, L, U};
+}
+
+std::pair<array, array> lu_factor(const array& a, StreamOrDevice s /* = {} */) {
+  validate_lu(a, s, "[linalg::lu_factor]");
+  auto out = lu_helper(a, s);
+  return std::make_pair(out[0], out[1]);
+}
+
+void validate_solve(
+    const array& a,
+    const array& b,
+    const StreamOrDevice& stream,
+    const std::string& fname) {
+  check_cpu_stream(stream, fname);
+  if (a.ndim() < 2) {
+    std::ostringstream msg;
+    msg << fname << " First input must have >= 2 dimensions. "
+        << "Received array with " << a.ndim() << " dimensions.";
+    throw std::invalid_argument(msg.str());
+  }
+
+  if (b.ndim() < 1) {
+    std::ostringstream msg;
+    msg << fname << " Second input must have >= 1 dimensions. "
+        << "Received array with " << b.ndim() << " dimensions.";
+    throw std::invalid_argument(msg.str());
+  }
+
+  if (a.shape(-1) != a.shape(-2)) {
+    std::ostringstream msg;
+    msg << fname << " First input must be a square matrix. "
+        << "Received array with shape " << a.shape() << ".";
+    throw std::invalid_argument(msg.str());
+  }
+
+  int lastDim = b.ndim() > 1 ? -2 : -1;
+  if (a.shape(-1) != b.shape(lastDim)) {
+    std::ostringstream msg;
+    msg << fname << " Last dimension of first input with shape " << a.shape()
+        << " must match second to last dimension of"
+        << " second input with shape " << b.shape() << ".";
+    throw std::invalid_argument(msg.str());
+  }
+
+  auto out_type = promote_types(a.dtype(), b.dtype());
+  if (out_type != float32) {
+    std::ostringstream msg;
+    msg << fname << " Input arrays must promote to float32. Received arrays "
+        << "with type " << a.dtype() << " and " << b.dtype() << ".";
+    throw std::invalid_argument(msg.str());
+  }
+}
+
+array solve(const array& a, const array& b, StreamOrDevice s /* = {} */) {
+  validate_solve(a, b, s, "[linalg::solve]");
+
+  // P, L, U matrices
+  const auto luf = lu(a, s);
+  auto perm = argsort(luf[0], -1, s);
+  int take_axis = -1;
+  if (b.ndim() >= 2) {
+    perm = expand_dims(perm, -1, s);
+    take_axis -= 1;
+  }
+  auto pb = take_along_axis(b, perm, take_axis);
+  auto y = solve_triangular(luf[1], pb, /* upper = */ false, s);
+  return solve_triangular(luf[2], y, /* upper = */ true, s);
+}
+
+array solve_triangular(
+    const array& a,
+    const array& b,
+    bool upper /* = false */,
+    StreamOrDevice s /* = {} */) {
+  validate_solve(a, b, s, "[linalg::solve_triangular]");
+  auto a_inv = tri_inv(a, upper, s);
+  return matmul(a_inv, b, s);
+}
+
 } // namespace mlx::core::linalg

mlx/linalg.h

@@ -74,6 +74,18 @@ array pinv(const array& a, StreamOrDevice s = {});
 
 array cholesky_inv(const array& a, bool upper = false, StreamOrDevice s = {});
 
+std::vector<array> lu(const array& a, StreamOrDevice s = {});
+
+std::pair<array, array> lu_factor(const array& a, StreamOrDevice s = {});
+
+array solve(const array& a, const array& b, StreamOrDevice s = {});
+
+array solve_triangular(
+    const array& a,
+    const array& b,
+    bool upper = false,
+    StreamOrDevice s = {});
+
 /**
  * Compute the cross product of two arrays along the given axis.
  */

mlx/primitives.h

@@ -2329,7 +2329,6 @@ class Eigh : public Primitive {
       : Primitive(stream),
         uplo_(std::move(uplo)),
         compute_eigenvectors_(compute_eigenvectors) {}
-
   void eval_cpu(const std::vector<array>& inputs, std::vector<array>& outputs)
       override;
   void eval_gpu(const std::vector<array>& inputs, std::vector<array>& outputs)
@@ -2350,4 +2349,16 @@ class Eigh : public Primitive {
   bool compute_eigenvectors_;
 };
 
+/* LU Factorization primitive. */
+class LUF : public Primitive {
+ public:
+  explicit LUF(Stream stream) : Primitive(stream) {}
+  void eval_cpu(const std::vector<array>& inputs, std::vector<array>& outputs)
+      override;
+  void eval_gpu(const std::vector<array>& inputs, std::vector<array>& outputs)
+      override;
+
+  DEFINE_PRINT(LUF)
+};
+
 } // namespace mlx::core