In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets S, T and U of a group G satisfying the Triple Product Property (TPP) as a means to bound the exponent $\omega$ of the matrix multiplication. We show that S, T and U may be be assumed to contain the identity and be otherwise disjoint. We also give a much shorter proof of the upper bound |S|+|T|+|U| <= |G|+2.
The naive algorithm for matrix multiplication is an O(n 3 ) algorithm. From Volker Strassen ([5]) we know that there is an O(n 2.81 ) algorithm for this problem. Winograd optimized Strassen's algorithm. While the Strassen-Winograd algorithm is the variant that is always implemented (for example in the famous GEMMW package), there are faster ones (in theory) that are impractical to implement. The fastest known algorithm runs in O(n 2.376 ) time (see [3] from Don Coppersmith and Shmuel Winograd). Most researchers believe that an optimal algorithm with O(n 2 ) runtime exists, but since 1987 no further progress was made in finding one.
Because modern architectures have complex memory hierarchies and increasing parallelism, performance has become a complex tradeoff, not just a simple matter of counting flops. Algorithms which make use of this technology were described in [1] by D’Alberto and Nicolau. An also well known method is Tiling: The normal algorithm can be speeded up by a factor of two by using a six loop implementation that blocks submatrices so that the data passes through the L1 Cache only once.
In 2003 Cohn and Umans introduced in [2] a group-theoretic approach to fast matrix multiplication. The main idea is to embed the matrix multiplication over a ring R into the group ring RG, where G is a (finite) group. A group G admits such an embedding, if there are subsets S, T and U which fulfill the so called Triple Product Property.
Definition (TPP). We say that the nonempty subsets S, T , and
We show that S, T and U may be be assumed to contain the identity and be otherwise disjoint.
Theorem 1. If S ′ , T ′ and U ′ fulfill the TPP, then there exists a triple S, T and U with
which also fulfills the TPP.
For the proof of our main result we need some auxiliary results.
Lemma 2. Let ∅ = X ⊆ G be a nonempty subset of a group G and g ∈ G.
Then
(3) |X| ≤ |Q(X)|.
Proof.
(1) Because X = ∅ there exists an x ∈ X and so 1 = xx -1 ∈ Q(X) follows.
(2) If g ∈ Q(X) then there are x, y ∈ X with g = xy -1 . This implies, that
(3) For a fixed x ∈ X the map X → Q(X), y → yx -1 is injective and therefore
Lemma 3. If S, T and U fulfill the TPP then
holds for all X = Y ∈ {S, T, U}.
Proof. We know 1 ∈ Q(X)∩Q(Y ) from Lemma 2(1). Now assume that |Q(X)∩Q(Y )| ≥ 2. In this case there is an 1 = x ∈ Q(X) ∩ Q(Y ). From Lemma 2(2) we know, that
x -1 ∈ Q(X)∩Q(Y ), too. Moreover 1 is an element of every right quotient and therefore the factors x, x -1 and 1 occur in {stu : Proof. Assume that |X ∩ Y | ≥ 2. Then there are x = y ∈ X ∩ Y . Therefore we have
. This is a contradiction to Lemma 3. Now we can prove our main result.
Proof of Theorem 1. We fix s 0 ∈ S ′ , t 0 ∈ T ′ and u 0 ∈ U ′ . Now we define S := {ss -1 0 : s ∈ S ′ } and T and U in the same way. Proof. The statement follows from Lemma 2(3) and Theorem 5.
Note that Theorem 5 is more effective than Corollary 6 when searching for TPP triples.
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