Tree-MNIST: Benchmark for Tree-Structured Data

• Tree-MNIST is a MNIST-style dataset and benchmark that integrates tree morphology and hierarchical semantics for vegetation classification.
• It employs an automated pipeline using CLIP features and deep Q-learning to filter, categorize, and balance low-resolution tree imagery.
• Related benchmarks like TreeSkel and Tree-SNE extend the concept to fine-grained tree skeletal images and multiscale tree embeddings.

to=arxiv_search.search 彩神争霸大发json code ： {"^^^^^^^^18query18 OR MNIST-Gen OR \18^^^^^^^^"The Tree-SNE Tree Exists^^^^^^^^18Tree-MNIST OR MNIST-Gen OR \18descending18^^^^^^^^" OR TreeSkel18^^^^^^^^", "^^^^^^^^18max_results18 ^^^^18Tree-MNIST OR MNIST-Gen OR \18query18^^^^, "^^^^^^^^18sort_by18 "^^^^^^^^18submittedDate18 "^^^^^^^^18sort_order18 "^^^^^^^^18descending18 to=arxiv_search.search კომენტary 天天中彩票在 code 微信天天中彩票՞ւ? to=arxiv_search.search wureg_json code ացին് ＿久久爱ाद्धी؟ {"^^^^^^^^18query18 OR MNIST-Gen OR \18^^^^^^^^"The Tree-SNE Tree Exists^^^^^^^^18Tree-MNIST OR MNIST-Gen OR \18descending18^^^^^^^^" OR TreeSkel18^^^^^^^^"," OR MNIST-Gen OR \18query18^^^^,"^^^^^^^^18sort_by18 to=arxiv_search.search เงินไทยฟรี code 手机天天中彩票? {"^^^^^^^^18query18 OR MNIST-Gen OR \18descending18^^^^^^^^"Tree-MNIST OR MNIST-Gen OR \18descending18^^^^^^^^""," to=arxiv.search 天天中彩票能_json code 的天天中彩票? {"^^^^^^^^18query18 to=arxiv_search 软件合法吗 code 天天中彩票篮球ి? {"^^^^^^^^18query18 Tree-MNIST denotes several related constructions in the cited arXiv literature rather than a single universally fixed object. Its most explicit and formalized meaning is a ^^^^18sort_by18^^^^-class, ^^^^18Tree-MNIST OR MNIST-Gen OR \18submittedDate18query18query18^^^^-image, PRESERVED_PLACEHOLDER^^^^18query18^^^^ grayscale dataset of trees generated by the MNIST-Gen pipeline, with labels Broadleaf Tree, Cactus, Coniferous Tree, and Palm (&&&^^^^18query18^^^^&&&). In adjacent usage, the expression also functions analogically for MNIST-like settings in which tree structure is central: TreeSkel, a fine-grained benchmark of PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^ tree skeleton images (&&&^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^&&&); graph-to-tree-to-image pipelines such as TreeRNN (&&&^^^^18 OR TreeSkel18^^^^&&&); and the Tree-SNE view of MNIST, where a one-parameter family of embeddings forms a PRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18^^^^-dimensional tree over scale (&&&^^^^18max_results18^^^^&&&).

^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^. Terminological scope

In the cited literature, the expression appears in several distinct senses. One is a concrete dataset intended for low-resolution tree classification. Another is a benchmark design pattern: a compact, structurally rich “X-MNIST” for tree-like objects. A third is a representational or geometric construction in which MNIST itself is organized as a tree.

Usage	Core object	Source
Tree-MNIST	^^^^18sort_by18^^^^-class PRESERVED_PLACEHOLDER_^^^^18max_results18^^^^ grayscale tree dataset	(&&&^^^^18query18^^^^&&&)
“Tree-MNIST” benchmark analogue	TreeSkel: PRESERVED_PLACEHOLDER_^^^^18sort_by18^^^^ binary tree skeletons	(&&&^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^&&&)
Tree-MNIST-style representation	Graph PRESERVED_PLACEHOLDER_^^^^18submittedDate18^^^^ tree PRESERVED_PLACEHOLDER_^^^^18sort_order18^^^^ image pipeline	(&&&^^^^18 OR TreeSkel18^^^^&&&)
Tree-MNIST as MNIST-through-scale	Tree-SNE tree of embeddings over PRESERVED_PLACEHOLDER_^^^^18descending18^^^^	(&&&^^^^18max_results18^^^^&&&)

The explicit dataset definition is the most literal usage. The other appearances are best understood as extensions of the same benchmark intuition: preserving MNIST-like compactness while introducing tree morphology, tree topology, or tree-valued multiscale structure.

^^^^18 OR TreeSkel18^^^^. Tree-MNIST as a generated vegetation dataset

Tree-MNIST is defined in MNIST-Gen as a ^^^^18sort_by18^^^^-class labeled dataset

PRESERVED_PLACEHOLDER_^^^^18query18^^^^

where the labels correspond to Broadleaf Tree, Cactus, Coniferous Tree, and Palm (&&&^^^^18query18^^^^&&&). The stated purpose is threefold: to provide a domain-specific MNIST-style benchmark for tree and vegetation classification; to demonstrate MNIST-Gen as a general pipeline for converting real-world hierarchical concepts into MNIST-style datasets; and to furnish a lightweight evaluation dataset when distributing a large, high-resolution tree dataset is impractical.

The dataset is built from a botanical hierarchy. The four main categories each contain three subcategories, giving ^^^^18Tree-MNIST OR MNIST-Gen OR \18 OR TreeSkel18^^^^ subcategories in total:

• Broadleaf Tree: Deciduous Broadleaf; Evergreen Broadleaf; Flowering Broadleaf.
• Cactus: Columnar Cactus; Barrel and Round Cactus; Branching and Pad Cactus.
• Coniferous Tree: Pine and Fir Trees; Spruce and Cedar; Juniper and Cypress.
• Palm: Fan Palm; Feather Palm; Coconut and Date Palm.

Each subcategory is associated with semantic characteristics. The cited examples include seasonal leaf drop, broad canopy, and autumn colors for deciduous broadleaf trees; tall vertical stems and ribbed surface for columnar cacti; needle leaves and conical or pyramidal shape for pine and fir trees; and radiating palmate fronds or pinnate leaves for distinct palm types (&&&^^^^18query18^^^^&&&). These semantic descriptors are not ornamental metadata: they are encoded in the pipeline and guide automatic categorization.

The data acquisition stage is keyword-driven. Images are obtained via queries to the Unsplash API, and the framework is also tied in general to Unsplash and Kaggle. For Tree-MNIST, around ^^^^18sort_by18query18query18^^^^ images per main class were collected before filtering and preprocessing. The final released dataset contains ^^^^18Tree-MNIST OR MNIST-Gen OR \18submittedDate18query18query18^^^^ labeled images total, with balanced representation across subcategories. The train/validation split is ^^^^18query18query18^^^^/^^^^18 OR TreeSkel18query18^^^^, giving approximately ^^^^18Tree-MNIST OR MNIST-Gen OR \18 OR TreeSkel18query18query18^^^^ training images and approximately ^^^^18max_results18query18query18^^^^ validation images (&&&^^^^18query18^^^^&&&).

^^^^18max_results18^^^^. Hierarchical semantics, reinforcement learning, and category-theoretic formulation

MNIST-Gen formalizes the hierarchy as

PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^

with PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18query18^^^^ the main categories, PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18Tree-MNIST OR MNIST-Gen OR \18^^^^ the subcategories under each PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18 OR TreeSkel18^^^^, and PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18max_results18^^^^ the semantic characteristics attached to each subcategory (&&&^^^^18query18^^^^&&&). For a raw image PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18sort_by18^^^^, the pipeline computes

PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18submittedDate18^^^^

where PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18sort_order18^^^^ is a CLIP image embedding from ViT-B/^^^^18max_results18 OR TreeSkel18^^^^, PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18descending18^^^^ contains brightness, contrast, and edge density, and PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18query18^^^^ records CLIP zero-shot object presence scores. Categorization is driven by the semantic score

PRESERVED_PLACEHOLDER_^^^^18Tree-MNIST OR MNIST-Gen OR \18Tree-MNIST OR MNIST-Gen OR \18^^^^

The label exposed in the final Tree-MNIST dataset is the parent main class of the selected subcategory. Thus the released task is ^^^^18sort_by18^^^^-way classification, but the generation process is informed by a ^^^^18Tree-MNIST OR MNIST-Gen OR \18 OR TreeSkel18^^^^-way internal semantic structure. This design is intended to preserve morphological diversity within each coarse class.

Selection and filtering are handled by a Deep Q-Learning agent. For each image, the agent chooses among three actions: keep in the predicted category, discard, or send to human review. The state includes CLIP features and semantic confidence, visual attributes, current class frequency for balancing, and similarity to already selected images to control redundancy. The reward is

PRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18query18^^^^

For Tree-MNIST specifically, the agent is described as discarding overexposed or overly similar samples based on semantic understanding while maintaining botanical diversity across subcategories, and as eliminating low-contrast or redundant samples (&&&^^^^18query18^^^^&&&).

The processing chain is additionally expressed as a composition of morphisms,

PRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18Tree-MNIST OR MNIST-Gen OR \18^^^^

with stages consisting of CLIP-based semantic analysis, hierarchical categorization, resize to fixed dimension, background removal via UPRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18 OR TreeSkel18^^^^-Net, center crop, grayscale conversion, and binarization or normalization. The paper casts raw image datasets and MNIST-style datasets as categories linked by a functor PRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18max_results18^^^^, with endofunctors describing augmentations and natural transformations comparing alternative pipelines (&&&^^^^18query18^^^^&&&). In Tree-MNIST, this formalism is used to state that the dataset is not merely a collection of resized images but the output of a composable semantic pipeline.

^^^^18sort_by18^^^^. Image format, evaluation protocol, and benchmark behavior

Tree-MNIST images are converted to MNIST-style grayscale at low resolution. The core preprocessing sequence is

PRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18sort_by18^^^^

The appendix also notes CLIP processing at PRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18submittedDate18^^^^, a configurable final resolution that can be PRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18sort_order18^^^^ or PRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18descending18^^^^, weighted grayscale conversion PRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18query18^^^^, and either Otsu’s method or an RL-learned threshold PRESERVED_PLACEHOLDER_^^^^18 OR TreeSkel18Tree-MNIST OR MNIST-Gen OR \18^^^^. Normalization to PRESERVED_PLACEHOLDER_^^^^18max_results18query18^^^^ with PRESERVED_PLACEHOLDER_^^^^18max_results18Tree-MNIST OR MNIST-Gen OR \18^^^^ is used for CNN training (&&&^^^^18query18^^^^&&&).

Benchmarking uses both a small CNN and classical machine-learning baselines on flattened PRESERVED_PLACEHOLDER_^^^^18max_results18 OR TreeSkel18^^^^ images. The CNN has two PRESERVED_PLACEHOLDER_^^^^18max_results18max_results18^^^^ convolution layers with ^^^^18Tree-MNIST OR MNIST-Gen OR \18sort_order18^^^^ and ^^^^18max_results18 OR TreeSkel18^^^^ filters, each followed by ReLU and max pooling, then a fully connected layer with ^^^^18sort_order18sort_by18^^^^ units and a softmax output over PRESERVED_PLACEHOLDER_^^^^18max_results18sort_by18^^^^ classes. Training uses Adam, learning rate PRESERVED_PLACEHOLDER_^^^^18max_results18submittedDate18^^^^, batch size ^^^^18max_results18 OR TreeSkel18^^^^, and ^^^^18Tree-MNIST OR MNIST-Gen OR \18query18^^^^ epochs (&&&^^^^18query18^^^^&&&).

On Tree-MNIST, the CNN attains ^^^^18query18sort_order18^^^^.^^^^18submittedDate18Tree-MNIST OR MNIST-Gen OR \18^^^^% accuracy, precision ^^^^18query18^^^^.^^^^18query18descending18sort_by18 OR TreeSkel18^^^^, recall ^^^^18query18^^^^.^^^^18query18sort_order18submittedDate18Tree-MNIST OR MNIST-Gen OR \18^^^^, and F^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^ score ^^^^18query18^^^^.^^^^18query18sort_order18Tree-MNIST OR MNIST-Gen OR \18max_results18^^^^. Random Forest reaches ^^^^18query18 OR TreeSkel18^^^^.^^^^18descending18max_results18^^^^% accuracy, Gradient Boosting ^^^^18query18Tree-MNIST OR MNIST-Gen OR \18^^^^.^^^^18Tree-MNIST OR MNIST-Gen OR \18submittedDate18^^^^%, K-Nearest Neighbors ^^^^18descending18Tree-MNIST OR MNIST-Gen OR \18^^^^.^^^^18query18sort_by18^^^^%, and the remaining reported baselines are lower, down to ^^^^18submittedDate18query18^^^^.^^^^18descending18sort_by18 for Gaussian Naive Bayes (&&&^^^^18query18^^^^&&&). The paper explicitly treats this as evidence that the task is nontrivial: even a relatively shallow CNN does not achieve the near-perfect accuracies typical on digit MNIST, because the tree images encode more complex morphology.

The reported confusion matrix shows strong diagonal values together with systematic errors. Broadleaf and Coniferous are confused in both directions, with ^^^^18 OR TreeSkel18submittedDate18^^^^ misclassifications each way. Palm is also confused with Cactus, including ^^^^18 OR TreeSkel18Tree-MNIST OR MNIST-Gen OR \18^^^^ Palm PRESERVED_PLACEHOLDER_^^^^18max_results18sort_order18^^^^ Cactus errors. The interpretation given is that silhouettes, partial views, and low-resolution crops can make some broadleaf and coniferous forms visually proximate, while fan palms and some columnar cacti can overlap morphologically under aggressive downsampling (&&&^^^^18query18^^^^&&&).

The authors further report that hierarchical semantic filtering combined with DQN reinforcement learning improves accuracy over random sampling by about ^^^^18submittedDate18^^^^–^^^^18query18^^^^ percentage points across datasets. A plausible implication is that Tree-MNIST’s difficulty is shaped not only by class semantics but also by curation quality: the benchmark is intended to be compact without being noisy.

^^^^18submittedDate18^^^^. TreeSkel and the fine-grained “Tree-MNIST” benchmark tradition

A different but closely related lineage is TreeSkel, introduced as a benchmark for fine-grained generative modelling of trees (&&&^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^&&&). TreeSkel is not the same object as the ^^^^18sort_by18^^^^-class Tree-MNIST dataset, but it is explicitly positioned relative to MNIST and described as serving the role that a “Tree-MNIST” would play for structurally complex tree shapes. Its content is ^^^^18 OR TreeSkel18^^^^D tree skeletons, i.e. medial axes of thin line drawings of trees, stored as PRESERVED_PLACEHOLDER_^^^^18max_results18descending18^^^^ binary images and later also treated as grayscale in generation. The dataset has ^^^^18Tree-MNIST OR MNIST-Gen OR \18submittedDate18^^^^,^^^^18query18query18query18^^^^ images, ^^^^18Tree-MNIST OR MNIST-Gen OR \18submittedDate18^^^^ tree species, and ^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^,^^^^18query18query18query18^^^^ images per class.

The ^^^^18Tree-MNIST OR MNIST-Gen OR \18submittedDate18^^^^ species are Acacia, Beech, Callistemon, Cedar, Chestnut, Elm, Japanese Maple, Kauri, Larch, Linden, Pine, Quaking Aspen, Small Maple, Teak, and White Birch. Data creation begins with artist-designed ^^^^18max_results18^^^^D tree models in Blender using the Sapling addon. Sapling parameters such as branching angles, numbers of branches, lengths, and curvature are randomly jittered around artist settings, producing ^^^^18 OR TreeSkel18submittedDate18query18^^^^ distinct ^^^^18max_results18^^^^D trees per species. Each ^^^^18max_results18^^^^D tree is then rendered from ^^^^18sort_by18^^^^ randomly chosen viewpoints, with the camera at average human height and the field of view chosen to keep the entire tree in frame. The binary silhouettes are skeletonized by medial-axis extraction, and the underlying ^^^^18max_results18^^^^D models are constrained so that the trunk splits at most twice (&&&^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^&&&).

TreeSkel is presented as harder than MNIST because trees impose global structural constraints, long-range dependencies, and fine-grained twig-level detail. The cited failure modes of models that work well on MNIST are disconnected components, blobs, spurious lines, and blurred fine structure. To address these issues, the paper proposes SketchGen, a VAE-GAN hybrid with an LSTM-based encoder and decoder with attention, a CNN discriminator, ^^^^18sort_order18sort_by18^^^^ time steps, hidden size ^^^^18 OR TreeSkel18submittedDate18sort_order18^^^^, latent dimensionality ^^^^18Tree-MNIST OR MNIST-Gen OR \18query18query18^^^^, a PRESERVED_PLACEHOLDER_^^^^18max_results18query18^^^^ read attention window, and a PRESERVED_PLACEHOLDER_^^^^18max_results18Tree-MNIST OR MNIST-Gen OR \18^^^^ write attention window (&&&^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^&&&).

Evaluation uses CNN-based false positive rates and a human perceptual study. SketchGen achieves ^^^^18sort_by18sort_order18^^^^.^^^^18sort_order18Tree-MNIST OR MNIST-Gen OR \18^^^^ in the artefacts study, ^^^^18submittedDate18query18^^^^.^^^^18 OR TreeSkel18sort_by18^^^^ in the realism study, and ^^^^18submittedDate18 OR TreeSkel18^^^^.^^^^18Tree-MNIST OR MNIST-Gen OR \18Tree-MNIST OR MNIST-Gen OR \18^^^^ in the user study, outperforming dcgan, vae-gan, and draw on this benchmark (&&&^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^&&&). In this strand of work, “Tree-MNIST” is best understood not as a single named dataset but as a benchmark ideal: a manageable image domain in which success requires both global tree connectivity and fine local detail.

^^^^18sort_order18^^^^. Tree-MNIST-style graph and tree representations

TreeRNN extends the same intuition from tree imagery to arbitrary graphs by first converting graphs into rooted BFS trees and then projecting those trees into image-like tensors suitable for conventional CNNs and RNNs (&&&^^^^18 OR TreeSkel18^^^^&&&). The paper explicitly frames this as being “exactly the spirit of a ‘Tree-MNIST’ framework”: non-image structured data are converted into fixed-size ^^^^18 OR TreeSkel18^^^^D arrays whose “pixels” encode node and edge features together with hierarchical relations.

For a graph PRESERVED_PLACEHOLDER_^^^^18sort_by18query18^^^^, the root is chosen as

PRESERVED_PLACEHOLDER_^^^^18sort_by18Tree-MNIST OR MNIST-Gen OR \18^^^^

where PRESERVED_PLACEHOLDER_^^^^18sort_by18 OR TreeSkel18^^^^ is the shortest-path distance from node PRESERVED_PLACEHOLDER_^^^^18sort_by18max_results18^^^^ to node PRESERVED_PLACEHOLDER_^^^^18sort_by18sort_by18^^^^. A BFS spanning tree then imposes direction from the center node to peripheral nodes. The subsequent tree-to-image projection maps each tree level to a row, allocates each node a block of consecutive columns whose width depends on subtree size, aligns children under their parent, and inserts padding to separate sibling groups. The resulting dataset-wide image size is

PRESERVED_PLACEHOLDER_^^^^18sort_by18submittedDate18^^^^

with PRESERVED_PLACEHOLDER_^^^^18sort_by18sort_order18^^^^ the maximum node count and PRESERVED_PLACEHOLDER_^^^^18sort_by18descending18^^^^ the maximum tree depth in the dataset (&&&^^^^18 OR TreeSkel18^^^^&&&).

Each pixel stores node features and the edge features connecting the node to its parent, after which an MLP with ^^^^18sort_order18sort_by18^^^^ hidden units and ReLU feeds a ^^^^18 OR TreeSkel18^^^^D RNN architecture, TreeRNN, followed by global max-pooling and a fully connected classifier. Reported results are competitive with state of the art: on MUTAG, TreeRNN reaches PRESERVED_PLACEHOLDER_^^^^18sort_by18query18^^^^; on PTC-MR, PRESERVED_PLACEHOLDER_^^^^18sort_by18Tree-MNIST OR MNIST-Gen OR \18^^^^; and on NCI^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^, PRESERVED_PLACEHOLDER_^^^^18submittedDate18query18^^^^ (&&&^^^^18 OR TreeSkel18^^^^&&&). The significance for Tree-MNIST is methodological rather than terminological. It suggests a reusable recipe for turning tree- or graph-structured objects into compact image benchmarks with explicit row/column semantics.

^^^^18descending18^^^^. Tree-MNIST as a multiscale tree of MNIST embeddings

A further and conceptually distinct usage appears in Tree-SNE. There, “Tree-MNIST” means MNIST seen through the additional scale dimension of Tree-SNE rather than through tree imagery (&&&^^^^18max_results18^^^^&&&). Instead of a single ^^^^18 OR TreeSkel18^^^^D t-SNE embedding, one obtains a one-parameter family of embeddings indexed by PRESERVED_PLACEHOLDER_^^^^18submittedDate18Tree-MNIST OR MNIST-Gen OR \18^^^^, stacked into a PRESERVED_PLACEHOLDER_^^^^18submittedDate18 OR TreeSkel18^^^^-dimensional object whose branches encode refinement from coarse digit classes to fine handwriting styles.

Tree-SNE uses the kernel family

PRESERVED_PLACEHOLDER_^^^^18submittedDate18max_results18^^^^

with PRESERVED_PLACEHOLDER_^^^^18submittedDate18sort_by18^^^^ recovering standard t-SNE, PRESERVED_PLACEHOLDER_^^^^18submittedDate18submittedDate18^^^^ giving the Gaussian kernel of SNE, and PRESERVED_PLACEHOLDER_^^^^18submittedDate18sort_order18^^^^ producing heavier tails. Embeddings are computed for a decreasing sequence PRESERVED_PLACEHOLDER_^^^^18submittedDate18descending18^^^^, with each layer initialized from the previous one, and then stacked along the PRESERVED_PLACEHOLDER_^^^^18submittedDate18query18^^^^-axis. The zero set of the gradient map

PRESERVED_PLACEHOLDER_^^^^18submittedDate18Tree-MNIST OR MNIST-Gen OR \18^^^^

defines the Tree-SNE tree geometrically (&&&^^^^18max_results18^^^^&&&).

The main theorem states that, at a generic point in this zero set, the stationary embeddings form a smooth properly embedded submanifold of dimension

PRESERVED_PLACEHOLDER_^^^^18sort_order18query18^^^^

with PRESERVED_PLACEHOLDER_^^^^18sort_order18Tree-MNIST OR MNIST-Gen OR \18^^^^ nonconstant on the manifold. The paper interprets the PRESERVED_PLACEHOLDER_^^^^18sort_order18 OR TreeSkel18^^^^ term as rigid-motion invariance and the remaining dimension as genuine variation in scale. Degenerate cases form a measure-zero set, so the tree exists generically (&&&^^^^18max_results18^^^^&&&).

Applied to MNIST, the result is a tree in which lower layers, corresponding to larger PRESERVED_PLACEHOLDER_^^^^18sort_order18max_results18^^^^, show thick digit-level branches, while smaller PRESERVED_PLACEHOLDER_^^^^18sort_order18sort_by18^^^^ yields progressively finer subbranches associated with handwriting styles. The cited description includes examples such as curved versus angular “^^^^18 OR TreeSkel18^^^^”, open versus closed “^^^^18sort_by18^^^^”, and different styles of “^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^”. In this sense, Tree-MNIST is neither a classification dataset nor a generative benchmark. It is a continuous multiscale visualization object in which MNIST’s cluster structure becomes explicitly tree-like.

Across these usages, Tree-MNIST names a shared research ambition: to preserve the compactness and experimental convenience associated with MNIST while replacing digit simplicity with tree morphology, tree topology, or tree-organized multiscale structure. The concrete ^^^^18sort_by18^^^^-class dataset of MNIST-Gen is the most literal realization of that ambition (&&&^^^^18query18^^^^&&&); TreeSkel and TreeRNN broaden it into benchmark and representation design (&&&^^^^18Tree-MNIST OR MNIST-Gen OR \18^^^^&&&, &&&^^^^18 OR TreeSkel18^^^^&&&); and Tree-SNE shows that MNIST itself can be transformed into a genuine tree over scale (&&&^^^^18max_results18^^^^&&&).