Support adding tensor of different orders about taco HOT 5 OPEN

Test case for the second semantics:

diff --git a/test/expr_storage-tests.cpp b/test/expr_storage-tests.cpp
index d7be97b..f164fac 100644
--- a/test/expr_storage-tests.cpp
+++ b/test/expr_storage-tests.cpp
@@ -185,6 +185,18 @@ INSTANTIATE_TEST_CASE_P(vector_scalar, expr,
                       }
                     },
                     {2.0, 4.0, 2.0, 2.0, 5.0}
+                    ),
+           TestData(Tensor<double>("a",{5},Format({Sparse})),
+                    {i},
+                    d5a("b",Format({Sparse}))(i) + da("c",Format())(),
+                    {
+                      {
+                        // Sparse index
+                        {0,4},
+                        {0,1,2,3,4}
+                      }
+                    },
+                    {2.0, 4.0, 2.0, 2.0, 5.0}
                     )
            )
 );

from taco.

This is how NumPy does it: https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html.

Matlab's latest release seems to have also adopted broadcasting: https://nickhigham.wordpress.com/2016/09/20/implicit-expansion-matlab-r2016b/

from taco.

If we adopt the first semantics, then would that not imply that the result of the computation can change depending on which storage format is used to store b, even if b is unchanged mathematically? That doesn't seem very intuitive or reasonable in my opinion.

from taco.

@stephenchouca that is a very good point and I think sufficient to adopt the second semantics.

@shoaibkamil The NumPy/Matlab documents don't seem to consider what happens if you're broadcasting over a sparse dimension, but it seems it is most natural broadcasting is over a dimension (semantics 2) and not over non-zero values (semantics 1). #6 also discusses broadcast semantics.

It seems like we all think the semantics 2 is correct.

from taco.

We should also support explicitly multiplying by a scalar/variable:

a(i) = 0.25 * b(i)

from taco.