mseeger commented on a change in pull request #15795: [Numpy] Differentiable svd
URL: https://github.com/apache/incubator-mxnet/pull/15795#discussion_r313779126
 
 

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 File path: src/operator/numpy/linalg/np_gesvd-inl.h
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+/*
+ * Licensed to the Apache Software Foundation (ASF) under one
+ * or more contributor license agreements.  See the NOTICE file
+ * distributed with this work for additional information
+ * regarding copyright ownership.  The ASF licenses this file
+ * to you under the Apache License, Version 2.0 (the
+ * "License"); you may not use this file except in compliance
+ * with the License.  You may obtain a copy of the License at
+ *
+ *   http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing,
+ * software distributed under the License is distributed on an
+ * "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+ * KIND, either express or implied.  See the License for the
+ * specific language governing permissions and limitations
+ * under the License.
+ */
+
+/*!
+ * Copyright (c) 2017 by Contributors
+ * \file np_gesvd-inl.h
+ * \brief Function definition of the SVD Operator.
+ */
+#ifndef MXNET_OPERATOR_NUMPY_LINALG_NP_GESVD_INL_H_
+#define MXNET_OPERATOR_NUMPY_LINALG_NP_GESVD_INL_H_
+
+#include <mxnet/operator_util.h>
+#include <vector>
+#include "../../tensor/la_op.h"
+#include "../../tensor/la_op-inl.h"
+
+namespace mxnet {
+namespace op {
+
+struct GesvdVecSign {
+  template<typename DType>
+  MSHADOW_XINLINE static void Map(int i, int m, int n, DType* UT,
+                                  DType* V, int ldut, int ldv) {
+    DType* vrow(V + i * ldv);
+    DType maxval(fabs(vrow[0])), vval(0.0);
+    int maxind(0);
+    for (int i = 1; i < n; ++i) {
+      vval = fabs(vrow[i]);
+      if (vval > maxval) {
+        maxval = vval;
+        maxind = i;
+      }
+    }
+    if (vrow[maxind] < 0) {
+      DType* utcol(UT + i % m + (i / m) * ldut * m);
+      for (int i = 0; i < n; ++i) {
+        vrow[i] = -vrow[i];
+        if (i < m) {
+          utcol[i * ldut] = -utcol[i * ldut];
+        }
+      }
+    }
+  }
+};
+
+// (UT, L, V) = gesvd(A) [singular value decomposition]
+// - V can overwrite A
+// - Needs workspace (both DType and int), size of which is determined by a
+//   workspace query
+struct gesvd {
+  template<typename xpu, typename DType>
+  static void op(const Tensor<xpu, 3, DType>& A,
+                 const Tensor<xpu, 3, DType>& UT,
+                 const Tensor<xpu, 2, DType>& L,
+                 const Tensor<xpu, 3, DType>& V,
+                 const OpContext& ctx,
+                 const nnvm::NodeAttrs& attrs) {
+    Stream<xpu> *s = ctx.get_stream<xpu>();
+    if (A.dptr_ != V.dptr_) Copy(V, A, s);
+    // From here on, we work on V only
+    // Reserve workspace (size determined by query)
+    int lwork(linalg_gesvd_workspace_query(UT[0], L[0], V[0], s));
+    Tensor<xpu, 1, DType> work = ctx.requested[0]
+      .get_space_typed<xpu, 1, DType>(Shape1(lwork), s);
+    // Loop over items in batch
+    for (index_t i = 0; i < UT.size(0); ++i) {
+      linalg_gesvd(UT[i], L[i], V[i], work, s);
+    }
+    // Set signs in a deterministic way
+    using namespace mxnet_op;
+    Kernel<GesvdVecSign, xpu>::Launch
+      (s, V.size(0) * V.size(1), V.size(1), V.size(2),
+       UT.dptr_, V.dptr_, UT.stride_, V.stride_);
+  }
+};
+
+// (A) => (UT, L, V)
+template<typename xpu, typename laop>
+void NumpyLaGesvdForward(const nnvm::NodeAttrs& attrs,
+                         const OpContext& ctx,
+                         const std::vector<TBlob>& inputs,
+                         const std::vector<OpReqType>& req,
+                         const std::vector<TBlob>& outputs) {
+  using namespace mshadow;
+  CHECK_EQ(inputs.size(), 1);
+  CHECK_EQ(outputs.size(), 3);
+  if (inputs[0].shape_.Size() == 0) {
+    return;
+  }
+  MSHADOW_SGL_DBL_TYPE_SWITCH(outputs[0].type_flag_, OType, {
+    mshadow::Stream<xpu> *s = ctx.get_stream<xpu>();
+    laop::op(inputs[0].FlatToKD<xpu, 3, OType>(s),
+             outputs[0].FlatToKD<xpu, 3, OType>(s),
+             outputs[1].FlatToKD<xpu, 2, OType>(s),
+             outputs[2].FlatToKD<xpu, 3, OType>(s), ctx, attrs);
+  });
+}
+
+// Helper for gesvd_backward. See technical report for details
+template<typename DType>
+DType gesvd_back_helper_eps(DType* X);
+
+template<>
+MSHADOW_XINLINE float gesvd_back_helper_eps(float* X) {
+  return 1e-30;
+}
+
+template<>
+MSHADOW_XINLINE double gesvd_back_helper_eps(double* X) {
+  return 1e-100;
+}
+
+struct GesvdBackHelper_dV {
+  template<typename DType>
+  MSHADOW_XINLINE static void Map(int k, int m, int n, DType* L, int ldl,
+                                  DType* dA, int ldda) {
+    const int offl(k * ldl);
+    const int offda(k * m * ldda);
+    DType denom(0.0);
+    const DType eps(gesvd_back_helper_eps(dA));
+    for (int i = 0; i < m; ++i) {
+      denom = L[offl + i];
+      if (denom < eps) denom = eps;
+      for (int j = 0; j < n; ++j) {
+        dA[offda + i * ldda + j] /= denom;
+      }
+    }
+  }
+};
+
+struct GesvdBackHelper_G1 {
+  template<typename DType>
+  MSHADOW_XINLINE static void Map(int k, int m, int n, DType* X, int ldx,
+                                  DType* L, int ldl) {
+    const int offl(k * ldl);
+    const int offx(k * m * ldx);
+    DType numer(0.0);
+    for (int i = 0; i < m; ++i) {
+      for (int j = 0; j < m; ++j) {
+        numer = L[offl + j];
+        X[offx + i * ldx + j] *= numer;
+      }
+    }
+  }
+};
+
+struct GesvdBackHelper_G2 {
+  template<typename DType>
+  MSHADOW_XINLINE static void Map(int k, int m, int n, DType* X, int ldx,
+                                  DType* L, int ldl, DType* dL, int lddl,
+                                  DType* dA, int ldda, DType* V, int ldv) {
+    const int offx(k * m * ldx);
+    const int offl(k * ldl);
+    const int offdl(k * lddl);
+    const int offda(k * m * ldda);
+    const int offv(k * m * ldv);
+    const DType eps(gesvd_back_helper_eps(X));
+    DType denom1(0.0), denom2(0.0), elem(0.0);
+
+    for (int i = 0; i < m - 1; ++i) {
+      for (int j = i + 1; j < m; ++j) {
+        denom1 = L[offl + i] - L[offl + j];
+        denom2 = L[offl + i] + L[offl + j];
+        if (denom1 < eps) denom1 = eps;
+        if (denom2 < eps) denom2 = eps;
+        elem = (X[offx + i * ldx + j] - X[offx + j * ldx + i]) / denom1 / 
denom2;
+        X[offx + i * ldx + j] = elem * L[offl + j];
+        X[offx + j * ldx + i] = elem * L[offl + i];
+      }
+    }
+    for (int i = 0; i < m; ++i) {
+      elem = DType(0.0);
+      for (int j = 0; j < n; ++j) {
+        elem += dA[offda + i * ldda + j] * V[offv + i * ldv + j];
+      }
+      elem = -elem + dL[offdl + i];
+      X[offx + i * ldx + i] = elem;
+    }
+  }
+};
+
+struct gesvd_backward {
+  template<typename xpu, typename DType>
+  static void op(const Tensor<xpu, 3, DType>& dUT,
+                 const Tensor<xpu, 2, DType>& dL,
+                 const Tensor<xpu, 3, DType>& dV,
+                 const Tensor<xpu, 3, DType>& UT,
+                 const Tensor<xpu, 2, DType>& L,
+                 const Tensor<xpu, 3, DType>& V,
+                 const Tensor<xpu, 3, DType>& dA,
+                 const Tensor<xpu, 3, DType>& tempMs,
+                 const Tensor<xpu, 3, DType>& tempMr,
 
 Review comment:
   Please ask if this is still unclear (but your comment is what I have in 
mind). You can mask out blocks of dA simply by moving the pointer, everything 
else (stride, etc) remains the same, because the n-axis is the continuous one. 
The final block will be (m, m2), where m2 <= m, but that should also not be a 
problem.

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