Question: Will enhancing diversity among the ro of VT improve the performance of matrix VT?
Evidence: BIBREF13 proposed a MP framework under which many of the recently introduced GNNs can be reformulated. MP consists in an aggregation phase followed by a combination phase BIBREF14. More precisely, let $G(V,E)$ be a graph, and let us consider $v \in V$. At time $t+1$, a message vector $\mathbf {m}_v^{t+1}$ is computed from the representations of the neighbors $\mathcal {N}(v)$ of $v$:

The new representation $\mathbf {h}^{t+1}_v$ of $v$ is then computed by combining its current feature vector $\mathbf {h}^{t}_v$ with the message vector $\mathbf {m}_v^{t+1}$:

Messages are passed for $T$ time steps. Each step is implemented by a different layer of the MP network. Hence, iterations correspond to network depth. The final feature vector $\mathbf {h}_v^T$ of $v$ is based on messages propagated from all the nodes in the subtree of height $T$ rooted at $v$. It captures both the topology of the neighborhood of $v$ and the distribution of the vertex representations in it.
If a graph-level feature vector is needed, e.g., for classification or regression, a READOUT pooling function, that must be invariant to permutations, is applied:

Next, we present the MP network we developed for document understanding.
We represent a document as a statistical word co-occurrence network BIBREF18, BIBREF19 with a sliding window of size 2 overspanning sentences. Let us denote that graph $G(V,E)$. Each unique word in the preprocessed document is represented by a node in $G$, and an edge is added between two nodes if they are found together in at least one instantiation of the window. $G$ is directed and weighted: edge directions and weights respectively capture text flow and co-occurrence counts.
$G$ is a compact representation of its document. In $G$, immediate neighbors are consecutive words in the same sentence. That is, paths of length 2 correspond to bigrams. Paths of length more than 2 can correspond either to traditional $n$-grams or to relaxed $n$-grams, that is, words that never appear in the same sentence but co-occur with the same word(s). Such nodes are linked through common neighbors.
Master node. Inspired by BIBREF3, our $G$ also includes a special document node, linked to all other nodes via unit weight bi-directional edges. In what follows, let us denote by $n$ the number of nodes in $G$, including the master node.
We formulate our AGGREGATE function as:

where $\mathbf {H}^t \in \mathbb {R}^{n \times d}$ contains node features ($d$ is a hyperparameter), and $\mathbf {A} \in \mathbb {R}^{n \times n}$ is the adjacency matrix of $G$. Since $G$ is directed, $\mathbf {A}$ is asymmetric. Also, $\mathbf {A}$ has zero diagonal as we choose not to consider the feature of the node itself, only that of its incoming neighbors, when updating its representation. Since $G$ is weighted, the $i^{th}$ row of $A$ contains the weights of the edges incoming on node $v_i$. $\mathbf {D} \in \mathbb {R}^{n \times n}$ is the diagonal in-degree matrix of $G$. MLP denotes a multi-layer perceptron, and $\mathbf {M}^{t+1} \in \mathbb {R}^{n \times d}$ is the message matrix.
The use of a MLP was motivated by the observation that for graph classification, MP neural nets with 1-layer perceptrons are inferior to their MLP counterparts BIBREF14. Indeed, 1-layer perceptrons are not universal approximators of multiset functions. Note that like in BIBREF14, we use a different MLP at each layer.
Renormalization. The rows of $\mathbf {D}^{-1}\mathbf {A}$ sum to 1. This is equivalent to the renormalization trick of BIBREF9, but using only the in-degrees. That is, instead of computing a weighted sum of the incoming neighbors' feature vectors, we compute a weighted average of them. The coefficients are proportional to the strength of co-occurrence between words. One should note that by averaging, we lose the ability to distinguish between different neighborhood structures in some special cases, that is, we lose injectivity. Such cases include neighborhoods in which all nodes have the same representations, and neighborhoods of different sizes containing various representations in equal proportions BIBREF14. As suggested by the results of an ablation experiment, averaging is better than summing in our application (see subsection SECREF30). Note that instead of simply summing/averaging, we also tried using GAT-like attention BIBREF11 in early experiments, without obtaining better results.
As far as our COMBINE function, we use the Gated Recurrent Unit BIBREF20, BIBREF21:

Omitting biases for readability, we have:

where the $\mathbf {W}$ and $\mathbf {U}$ matrices are trainable weight matrices not shared across time steps, $\sigma (\mathbf {x}) = 1/(1+\exp (-\mathbf {x}))$ is the sigmoid function, and $\mathbf {R}$ and $\mathbf {Z}$ are the parameters of the reset and update gates. The reset gate controls the amount of information from the previous time step (in $\mathbf {H}^t$) that should propagate to the candidate representations, $\tilde{\mathbf {H}}^{t+1}$. The new representations $\mathbf {H}^{t+1}$ are finally obtained by linearly interpolating between the previous and the candidate ones, using the coefficients returned by the update gate.
Interpretation. Updating node representations through a GRU should in principle allow nodes to encode a combination of local and global signals (low and high values of $t$, resp.), by allowing them to remember about past iterations. In addition, we also explicitly consider node representations at all iterations when reading out (see Eq. DISPLAY_FORM18).

After passing messages and performing updates for $T$ iterations, we obtain a matrix $\mathbf {H}^T \in \mathbb {R}^{n \times d}$ containing the final vertex representations. Let $\hat{G}$ be graph $G$ without the special document node, and matrix $\mathbf {\hat{H}}^T \in \mathbb {R}^{(n-1) \times d}$ be the corresponding representation matrix (i.e., $\mathbf {H}^T$ without the row of the document node).
We use as our READOUT function the concatenation of self-attention applied to $\mathbf {\hat{H}}^T$ with the final document node representation. More precisely, we apply a global self-attention mechanism BIBREF22 to the rows of $\mathbf {\hat{H}}^T$. As shown in Eq. DISPLAY_FORM17, $\mathbf {\hat{H}}^T$ is first passed to a dense layer parameterized by matrix $\mathbf {W}_A^T \in \mathbb {R}^{d \times d}$. An alignment vector $\mathbf {a}$ is then derived by comparing, via dot products, the rows of the output of the dense layer $\mathbf {Y}^T \in \mathbb {R}^{(n-1) \times d}$ with a trainable vector $\mathbf {v}^T \in \mathbb {R}^d$ (initialized randomly) and normalizing with a softmax. The normalized alignment coefficients are finally used to compute the attentional vector $\mathbf {u}^T \in \mathbb {R}^d$ as a weighted sum of the final representations $\mathbf {\hat{H}}^T$.


Note that we tried with multiple context vectors, i.e., with a matrix $\mathbf {V}^T$ instead of a vector $\mathbf {v}^T$, like in BIBREF22, but results were not convincing, even when adding a regularization term to the loss to favor diversity among the rows of $\mathbf {V}^T$.
Master node skip connection. $\mathbf {h}_G^T \in \mathbb {R}^{2d}$ is obtained by concatenating $\mathbf {u}^T$ and the final master node representation. That is, the master node vector bypasses the attention mechanism. This is equivalent to a skip or shortcut connection BIBREF23. The reason behind this choice is that we expect the special document node to learn a high-level summary about the document, such as its size, vocabulary, etc. (more details are given in subsection SECREF30). Therefore, by making the master node bypass the attention layer, we directly inject global information about the document into its final representation.
Multi-readout. BIBREF14, inspired by Jumping Knowledge Networks BIBREF12, recommend to not only use the final representations when performing readout, but also that of the earlier steps. Indeed, as one iterates, node features capture more and more global information. However, retaining more local, intermediary information might be useful too. Thus, instead of applying the readout function only to $t=T$, we apply it to all time steps and concatenate the results, finally obtaining $\mathbf {h}_G \in \mathbb {R}^{T \times 2d}$ :

In effect, with this modification, we take into account features based on information aggregated from subtrees of different heights (from 1 to $T$), corresponding to local and global features.
Through the successive MP iterations, it could be argued that MPAD implicitly captures some soft notion of the hierarchical structure of documents (words $\rightarrow $ bigrams $\rightarrow $ compositions of bigrams, etc.). However, it might be beneficial to explicitly capture document hierarchy. Hierarchical architectures have brought significant improvements to many NLP tasks, such as language modeling and generation BIBREF24, BIBREF25, sentiment and topic classification BIBREF26, BIBREF27, and spoken language understanding BIBREF28, BIBREF29. Inspired by this line of research, we propose several hierarchical variants of MPAD, detailed in what follows. In all of them, we represent each sentence in the document as a word co-occurrence network, and obtain an embedding for it by applying MPAD as previously described.
MPAD-sentence-att. Here, the sentence embeddings are simply combined through self-attention.
MPAD-clique. In this variant, we build a complete graph where each node represents a sentence. We then feed that graph to MPAD, where the feature vectors of the nodes are initialized with the sentence embeddings previously obtained.
MPAD-path. This variant is similar to the clique one, except that instead of a complete graph, we build a path according to the natural flow of the text. That is, two nodes are linked by a directed edge if the two sentences they represent follow each other in the document.
Note that we tried with multiple context vectors, i.e., with matrix VT instead of a vector vT, like in (Lin et al. 2017), but results were not convincing, even when adding a reg larization term to the loss to favor diversity among the ro of VT.
Answer: No, they tried but the results were convincing.