Background: Medical image classification using deep learning is a critical task, yet its effectiveness is constrained by the scarcity of labeled data, which is expensive to acquire. Semi-supervised learning (SSL) methods address this by leveraging unlabeled data. Common autoencoder (AE)-based approaches use reconstruction as a training signal. However, standard reconstruction loss minimization does not guarantee that the resulting latent space will be optimally structured for the classification task, as the model may focus on diagnostically irrelevant features. Objective: To develop and experimentally validate a novel latent space regularization method: fractal-driven regularization (FDR). The goal is to improve classification metrics for medical images under conditions of severe labeled data scarcity (5%) by integrating fractal dimension (FD) as an additional, a priori training signal. Methods: The proposed model (FDR-AE) is based on an autoencoder architecture, augmented with two heads attached to the latent space: a classification head and a regression head. The regression head is trained to predict the input image's FD, which is pre-calculated using the "box-counting" method. The total loss function is a combination of three components: classification loss (on 5% labeled data) and both reconstruction and fractal regression losses (on 100% of data). The method's efficacy was validated on three datasets of different modalities (ISIC2024, COVID-19 Radiology, Brain Tumor MRI), comparing it against a baseline convolutional network (Base-CNN) and a standard semi-supervised autoencoder (SSL-AE). Results: The experiments demonstrated a consistent advantage for the proposed method. On the ISIC2024 dataset, FDR-AE achieved an F1-Score of 0.508 for the "malignant" class, compared to 0.431 for SSL-AE and 0.304 for Base-CNN. On the COVID-19 dataset, the F1-Score for the "covid19" class was 0.722 for FDR-AE versus 0.695 for SSL-AE. In the 4-class Brain Tumor task, FDR-AE showed improved F1-Scores across all classes, with the most significant gains (+0.079 and +0.054) observed for classes 0 and 3, which also had the greatest mutual statistical difference in their FD values. Conclusions: Fractal-driven regularization demonstrates that FD is a valuable a priori signal for learning higher-quality, structurally-grounded representations in SSL tasks. The method is particularly effective on simple architectures under severe data scarcity. Prospects for future research include using FDR as a pre-training method or implementing a dynamic coefficient for the regression component of the loss function.
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