About VAlrind

VAlrind is a Python Library based on Numpy for Training Neural Networks, and Training Graph Neural Networks There are many SOTA Libraries such as Pytorch/Tensorflow/HuggingFace and much more for doing so.

But VAlrind have a different approach with NNs.

Working

It encodes the values of a Matrix of (m x n) into a Row/Col of len(row) = m/len(col) = n. Then it uses Bivariate Functions to calculate the values of Matrix at every intersection of row/col.

It calculates the row/col of the Final Matrix (after the Matrix Multiplication), directly from the row/col of the Constituent Matrices.

Doing so decreases the Time Complexity of Matrix Multiplication to O(m + n).

In this Digram we have used Multiplication $f(x , y) = x * y$ as the Bivariate Function

Mathametics

For the Multiplication

We denote the row/col of the matrices as f_row/f_col/s_row/s_col/o_row/o_col

f_row	Row of First Matrix
f_col	Col of First Matrix
s_row	Row of Second Matrix
s_col	Col of Second Matrix
o_row	Row of Output Matrix
o_col	Col of Output Matrix

If we want to calculate what should be at the position [0 , 0] in the Output Matrix. The traditional way is to

sum(
    (f_col[0] * f_row) * 
    (s_row[0] * s_col)
) 

$$O_{0,0} = \sum((f_-col_0 * f_-row) * (s_-row_0 * s_-col))$$

We can break down the formula as

$$O_{0,0} = \sum(f_-col_0 * f_-row * s_-row_0 * s_-col)$$

As we try to iterate over the whole matrix

$$O_{0->n,0->m} = \sum(f_-col_{0->n} * f_-row * s_-row_{0->m} * s_-col)$$

As f_row and s_col act as constants we can take them out of the summission

$$O_{0->n,0->m} = f_-row * s_-col\sum(f_-col_{0->n} * s_-row_{0->m})$$

We introduce some denotions here as

$$f_-row -> o_-row$$

$$s_-col\sum(f_-col_{0->n} * s_-row_{0->m}) -> o_-col$$

Further denoting this formula as

$$O = o_-row * o_-col$$

And thus making our formula for the output matrix.

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For any Bivariate Function

We denote any Bivariate Function to be (π) and for this Bivariate Function

$$O_{0,0} = \sum((f_-col_0 (π) f_-row) * (s_-row_0 (π) s_-col))$$

$$=>O_{0->n,0->m} = \sum(f_-col_{0->n} (π) f_-row * s_-row_{0->m} (π) s_-col)$$

$$ => O_{0->n,0->m} = f_-row (π) s_-col\sum(f_-col_{0->n} * s_-row_{0->m})$$

We introduce some denotions here as

$$f_-row -> o_-row$$

$$s_-col\sum(f_-col_{0->n} * s_-row_{0->m}) -> o_-col$$

Further denoting this formula as

$$O = o_-row (π) o_-col$$

And thus making our formula for the output matrix.

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Note : Sorry for the inconvinence, because of some ambiguity in account with PyPi, we are not able to fully publish the Library, this will be fixed soon

For the Mean-time use the comannd below in your terminal to install the Library

pip install git+https://github.com/avalrind/VAl_rind