We can adjust the RSA Hash Accumulator to store a matrix of values. This way we can store more values using less primes.
The idea behind the accumulator is that every hashed value is multiplied with every prime except one. To proof a value is in the accumulator we can group all other values together, because they are all multiplied by the same prime that is missing from the value we like to proof.
To store a matrix instead of a vector, we will multiply all hashed values with every prime except two, this way we can store n*m values using only n+m primes. The downside of this approach is that we need two 2048 bit witnesses to proof inclusion of one value.
Hash accumulator for a mxn matrix
Let x_{i,j} be the values of the matrix we like to store.
Let p_{1},..,p_{n} and q_{1},..,q_{m} be the n+m distinct large primes.
We define P_S and Q_S to be the product of all their respective primes not contained in the set S (not g raised to the power as above)
P_S = \prod\limits_{\substack{k=1,k\not\in S}}^n p_{k}
Q_S = \prod\limits_{\substack{k=1,k\not\in S}}^m q_{k}
The accumulator A = g^{\sum\limits_{k=1}^m \sum\limits_{l=1}^n Q_{k}P_{l}\textit{h}(x_{k,l})} \mod N
To proof x_{i,j} is stored in spot i,j, we need two witnesses:
W_p = g^{\sum\limits_{l=1,l \neq j}^n Q_{i}P_{l,j}\textit{h}(x_{i,l})} \mod N
W_q = g^{\sum\limits_{k=1,k\neq i}^m \sum\limits_{l=1}^n Q_{k,i}P_{l}\textit{h}(x_{k,l})} \mod N
The verifier has to check: g^{Q_iP_j{\textit{h}(x_{i,j})}} W_q^{q_i}W_p^{p_j} \equiv A \mod N
If we select a set of columns and a set of rows we can proof in a similar manner, all values that are both in the row set as in the column set, using just two 2048 bit witnesses.
If we like, this can be extended to store n-dimensional matrices.
Edit: adding a small example to make it less abstract
A small example for a 2x3 matrix
We are using three p primes p_1, p_2 and p_3 for the columns and two q primes q_1 and q_2 for the rows.
We can store six values: x_{1,1}, x_{1,2}, x_{1,3}, x_{2,1}, x_{2,2}, x_{2,3}
The accumulator A=g^{q_2p_2p_3\textit{h}(x_{1,1}) + q_2p_1p_3\textit{h}(x_{1,2}) + q_2p_1p_2\textit{h}(x_{1,3}) + q_1p_2p_3\textit{h}(x_{2,1}) + q_1p_1p_3\textit{h}(x_{2,2}) + q_1p_1p_2\textit{h}(x_{2,3})}
To proof x_{2,2} is in the accumulator we need witnesses:
W_p = g^{q_1p_3\textit{h}(x_{2,1}) + q_1p_1\textit{h}(x_{2,3})}
W_q = g^{p_2p_3\textit{h}(x_{1,1}) + p_1p_3\textit{h}(x_{1,2}) + p_1p_2\textit{h}(x_{1,3})}
The verifier has to check: g^{q_1p_1p_3\textit{h}(x_{2,2})}W_q^{q_2}W_p^{p_2} \equiv A \mod N