Training shallow neural network (no hidden layer) for MNIST classification. Getting low accuracy

I am trying to implement classification of MNIST dataset without any hidden layers. So the data is input as 784x1 vector and output is 10x1 vector (after one-hot encoding).

The problem is that once I train the network, I am getting very low accuracy (~1%). The reason for that is unclear to me. My guess is that my update rules are incorrect. After training, the output vector on test data (a_test below) is not one-hot anymore but has multiple 1's. I am not able to figure out where I am going wrong.

alp       = 0.0001;   % learning rate
epoch     = 50;       % number of iterations
dim_train = 784;      % dimension of input data vector 
for itr = 1: epoch
    z = W*image_train+repmat(b',n_train,1)'; % image_train is a 784*60000 data matrix and W is 10*784 weight matrix and b is bias
    a = 1./(1+exp(-z)); 
    a = a+0.001;       % to avoid zero inside log when calculating cross entropy loss
    a_flg = sum(a);

    for i = 1:n_train
        a(:,i) = a(:,i)/a_flg(i);  % normalizing output
    end  
    L = -sum( Y_train.*log(a), 'all' );   % calculation loss
     dLdW = 1/dim_train*(a-Y_train)*image_train';   % calculating dL/dW
     dLdb = 1/dim_train*(a-Y_train)*ones(n_train,1);% calculating dL/db
     W = W - alp*dLdW;    % updating weights (gradient descent)
     b = b - alp*dLdb;    % updating bias    (gradient descent)
    
    loss(itr) = 1/n_train*L;
end
%% Testing
a_test = 1./(1+exp(-(W*image_test+repmat(b',n_test,1)')));
for i = 1:n_test
    a_test(~(a_test(:,i)==max(a_test(:,i))),i)=0;
end

Answer 1

[Srivardhan Gadila]

The network architecture defined without any hidden layer may not be able to learn to classify the digits beloging to 10 classes. It is reccommended to use few hidden layers.

You can also refer to Define Custom Deep Learning Layers & Create Simple Deep Learning Network for Classification