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More memory-efficient inner product for large last dimension?


I need to compute vector-vector, matrix-vector, and matrix-matrix
inner products, for vectors and matrices whose elements are not
scalars, but very large lists (~ 1.2M element each). I need Dot[] to
ignore the last tensor index, but it has no parameter for this, like
Outer's last "n_i " arguments. So I implemented my own. Unfortunately,
the matrix-vector and matrix-matrix products consume excessive amounts
of memory. The matrix-vector product peak memory footprint is ~ 800MB,
for ~ 110MB total input, and the matrix-matrix product peaks at ~ 1.8
GB for ~ 180MB input. Apparently, memory overhead is ~ 8-10X.

Here is a trace of system memory use (working set and its peak) for
the above Mathematica evaluations. My system is Windows 7, 2.5 GHz
Intel Core 2 Duo, 4GB RAM (Lenove R61 laptop). (PDF ~ 180KB)

Please find below my implementation of "dot", which ignores sublists
below level 1 or 2 (depends on product type). Can it be made more

Thank you,

Vince Virgilio

dot[l1_, l2_, 1] := l1*l2 // Total;

dot[l1_, l2_, 2, n2_: 1] :=
  ReleaseHold@Dot[Map[Hold, l1, {2}], Map[Hold, l2, {n2}]];

In[3]:= l1 = RandomReal[1., {3, 1200000}];
l2 = RandomReal[1., {3, 3, 1200000}];

In[5]:= ByteCount@l1

Out[5]= 28800128

In[6]:= ByteCount@l2

Out[6]= 86400132

(* Vector-Vector inner product *)

In[7]:= l3 = dot[l1, l1, 1];

Out[8]= {1200000}

(* Matrix-Vector inner product *)

l4 = dot[l2, l1, 2]; (* Memory use peaks @ ~ 800 MB *)

In[8]:= Dimensions@l4

Out[8]= {3, 1200000}

(* Matrix-Matrix inner product *)

l5 = dot[l2, l2, 2, 2]; (* Memory use peaks @ ~ 1.8 GB! *)

In[8]:= Dimensions@l5

Out[8]= {3, 3, 1200000}

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