\emph{Canonical partition} $\mathcal{P}$ crops the index-restricted d-hop neighborhood around the center node from the target graph. $\mathcal{D}(G_t,v_i,v_c)$ means the shortest distance between $v_i$ and $v_c$ on $G_t$.
 \begin{equation}
        \mathcal{P}(G_t, v_c, d) = G_c, \operatorname{ s.t. } G_c \subseteq G_t, V_c = \{ v_i \in V_t|\mathcal{D}(G_t,v_i,v_c) \leq d , v_i \leq v_c\}
\end{equation}

The graph $G_c$ obtained by canonical partition is called the \emph{canonical neighborhood}. Canonical neighborhoods can correctly substitute the target graph in canonical count. The subgraph count of query in target equals the summation of the canonical count of query in canonical neighborhoods for all target nodes. Canonical neighborhoods are acquired with canonical partition $\mathcal{P}$, given any $d$ greater than the diameter of the query.
\begin{equation}
	\mathcal{C}(G_q,G_t) = \sum_{v_c \in V_t} \mathcal{C}_c(G_q, \mathcal{P}(G_t, v_c, d),v_c), 
	d \geq \max_{v_i, v_j \in V_q} \mathcal{D}(G_q, v_i, v_j)
\end{equation}

A **Harmonic Block** is an image model component that utilizes [Discrete Cosine Transform](https://paperswithcode.com/method/discrete-cosine-transform) (DCT) filters. Convolutional neural networks (CNNs) learn filters in order to capture local correlation patterns in feature space. In contrast, DCT has preset spectral filters, which can be better for compressing information (due to the presence of redundancy in the spectral domain).

DCT has been successfully used for JPEG encoding to transform image blocks into spectral representations to capture the most information with a small number of coefficients. Harmonic blocks learn how to optimally combine spectral coefficients at every layer to produce a fixed size representation defined as a weighted sum of responses to DCT filters. The use of DCT filters allows to address the task of model compression.

Harmonic Block

Harmonic Convolutional Networks based on Discrete Cosine Transform

Canonical Partition

DeSCo: Towards Generalizable and Scalable Deep Subgraph Counting

**Fast AutoAugment** is an image data augmentation algorithm that finds effective augmentation policies via a search strategy based on density matching, motivated by Bayesian DA. The strategy is to improve the generalization performance of a given network by learning the augmentation policies which treat augmented data as missing data points of training data. However, different from Bayesian DA, the proposed method recovers those missing data points by the exploitation-and-exploration of a family of inference-time augmentations via Bayesian optimization in the policy search phase. This is realized by using an efficient density matching algorithm that does not require any back-propagation for network training for each policy evaluation.

Source	DeSCo: Towards Generalizable and Scalable Deep Subgraph Counting
Year	2000
Data Source	CC BY-SA - https://paperswithcode.com