Efficient and interpretable DNA/RNA representation using Komlós–Hadamard transforms

Abstract

This study introduces a novel encoding scheme for DNA/RNA sequences, integrating Komlós and Hadamard transforms. Unlike traditional One-Hot encoding, this approach offers a more informative representation of omics data while significantly reducing computational complexity. However, it is important to note that the Komlós transform component provides fewer features and does not utilize sparse codes. By leveraging the inherent properties of these transforms, our method effectively captures complex patterns within the data, leading to improved model accuracy and reduced training times. When combined with an image transformation, this encoding scheme demonstrates particularly efficient results, achieving superior performance across various predictive tasks with significantly lower computational resource demands compared to One-Hot encoding. Our findings suggest that this novel encoding scheme, particularly when integrated with Hilbert Curve mapping or sequence to image analysis, holds significant promise for advancing DNA/RNA data analysis by offering a more efficient and effective approach to feature representation.

Introduction

DNA can be seen as a sequence composed of the letters A, C, G, and T, representing the nucleotides that form its alphabet. This sequence serves as the fundamental code and blueprint for all cellular processes. However, DNA is more than just a linear sequence, but is a molecule with a distinct three-dimensional structure. This spatial organization is facilitated by histone proteins, which interact with DNA to help it achieve its proper configuration. Together, DNA and these supporting proteins form chromatin, whose structure ultimately influences the functional characteristics of specific DNA regions [1, 2].

Accurate encoding of the DNA sequence shall enable the analysis of genomic variations, such as single nucleotide polymorphisms and structural variants, which are crucial for understanding disease mechanisms and evolutionary processes. Furthermore, DNA sequence encoding is indispensable for studying gene function through transcriptomics, proteomics, and epigenetics, providing critical insights into gene regulation, interactions, and expression patterns [3].

However, the vast amount and complexity of DNA sequence data pose significant challenges, including data management, integration with other omics datasets, and ethical considerations related to privacy and data sharing. Seminal studies in this field have laid the groundwork for addressing these challenges, presenting novel algorithms, ethical frameworks, and insights into the implications of DNA sequence encoding in genomics research.

Currently, machine learning and artificial intelligence algorithms still rely on One-Hot encoding as the standard for DNA sequence representation in many applications [4–7] where it is extensively employed to represent DNA sequences, both in their raw form for short-range genomic modeling and when combined with other approaches such as Hilbert curve spatial transformations for long-range sequence analysis, as well as in DNA language models that process nucleotide patterns as biological ’text’.

From a statistical analysis perspective, handling One-Hot encoded data poses two main challenges. First, it dilutes information, making it difficult for models to learn from rare categories. Second, representing each category as a separate dimension results in high-dimensional feature vectors, leading to increased computational and memory demands and making the learning task more complex, which in turn affects statistical estimation quality [8]. Although dimensionality reduction techniques can mitigate this issue, they risk losing critical information.

Moreover, encoding each unique entry as an orthogonal vector ignores any similarity or overlap inherent in their string representations. One-Hot encoding also fails to generalize to new categories encountered during testing, as it cannot assign meaningful vectors to unseen inputs. Using heuristics like assigning a zero vector for new categories can cause collisions when multiple new categories arise. Consequently, One-Hot encoding is unsuitable for online learning scenarios, as the emergence of new categories requires re-encoding the entire dataset, leading to unbounded growth in feature dimensionality.

Approaches have been explored but remain fundamentally limited. Gödel numbers [9] and k-mer countings are two prominent examples. Gödel encoding suffers from exponential growth, making it computationally intractable and difficult to store while k-mer methods face challenges in capturing sequence variability [9]. Specifically, short sequences may lack sufficient informative k-mers, whereas long sequences generate an excessive number, leading to severe data imbalance. Consequently, selecting an optimal k-mer length becomes a critical trade-off in omics representation–balancing sensitivity (to capture true overlaps despite sequencing errors) against specificity (to resolve genomic repeats) [10–12].

FastDNA encodes sequences by treating k-mers as “words” and learning vector embeddings similar to natural language processing, which allows efficient and scalable classification of genomic and metagenomic data [10]. The performance of FastDNA depends heavily on the choice of k which related is to the subsequences (k-mers) length.

Nucleotide Chemical Property (NCP) [13] maps each nucleotide into a three-dimensional vector based on its inherent chemical characteristics thereby providing biologically relevant features. NCP suffers from loss of information and poor discrimination. Nucleotide Chemical Property And Frequency Encoding (NCPAFE), an extention of NCP, is a fusion of NCP with structural information, transforms raw nucleotide sequences into numerical formats that reflect both their biological characteristics and their frequency of occurrence [14]. Binary encoding represents nucleotides (A, C, G, T) using a fixed-length binary code instead of sparse One-Hot vectors (for a 2-bit binary encoding scheme, one can have A = 00, C = 01, G = 10, T = 11). This may introduce artificial similarity between nucleotides and may impose an arbitrary ordering not grounded in biological properties [15]. Finally, Enhanced Nucleic Acid Composition (ENAC) calculates nucleotide frequencies within fixed-length sliding windows along the sequence, enabling the capture of local compositional patterns but it is inadequate for modeling global or long-range sequence relationships [16].

In this paper, we propose a novel coding method noted "Komlós" for representing biological sequences that overcomes the limitations of One-Hot encoding and the previously stated approaches. Furthermore Komlós is designed to: (1) reduce sparsity and memory usage, (2) avoid exponential growth as in Gödel encodings, ensuring scalability for long sequences, (3) is universally applicable to DNA, RNA, lnRNA etc.., enabling consistent representation across genomics tracks.

Related work

DNA sequence representation

K-mer embedding approaches are essential for transforming genomic sequences into numerical formats suitable for machine learning. Traditional deep learning models such as DeepSEA [17] pioneered the use of One-Hot encoded k-mers as input to convolutional neural networks (CNNs), enabling accurate predictions of chromatin effects and regulatory elements by identifying local sequence motifs. Quang et al. extended this framework by incorporating a hybrid CNN–recurrent architecture, adding bidirectional LSTMs to capture long-range dependencies and complex regulatory syntax across the DNA [18]. These models rely heavily on sparse One-Hot encodings, which, although expressive, are computationally intensive. More recent work has explored Hilbert curve-based embeddings where DNA sequences are spatially mapped into 2D grids that preserve k-mer locality [19].

Collectively, these embedding strategies have advanced tasks like enhancer prediction, variant effect modeling, and chromatin state classification. However, their reliance on dense encodings or high-order transformations motivates the need for more compact, biologically-informed representations–such as the Komlós-based encoding introduced in this study.

To accurately represent all k-mers in a DNA sequence, a vector of length is required, where is the length of each word, and 4 corresponds to the nucleotide alphabet A, C, G, T. For instance, when considering 1-mers, a One-Hot vector of length 4 can be used, with A represented as [1, 0, 0, 0], C as [0, 1, 0, 0], G as [0, 0, 1, 0], and T as [0, 0, 0, 1]. In the case of a DNA sequence consisting of 4-mers, each 4-mer can be transformed into a One-Hot vector of length .

DNA sequence representation for Hilbert curve

Hilbert curve is a type of space-filling curve and a continuous fractal mapping that passes through every point in the unit square, effectively filling the two-dimensional space as its order approaches infinity [20]. For a Hilbert curve of order , it subdivides the unit square into a grid and the curve traverses each of the cells exactly once in a specific recursive pattern that preserves locality. When the spatial representation of a DNA sequence is used with Hilbert curve, one need to represent the sequence as a 2D grid, the n-length sequence is first converted into n-k+ 1 overlapping k-mers, which are then mapped onto a Hilbert curve of order l, corresponding to a grid. Each pixel in the grid is assigned a One-Hot encoded vector of length , representing all possible k-mers, resulting in a final image with dimensions , where the total pixels of the grid of dimension . For example, if the length of the sequence is and 3-mers is used, we obtain 64 pixels on the grid using the minimum such that (), each pixel containing a 64-dimensional One-Hot-encoded vector. We can then use a standard convolutional neural network to process this image representation. Alternatively, when using the same setup with Komlós encoding (our proposed approach), each nucleotide is represented by a compact 3-dimensional vector. For k-mers, the encoding produces a concatenated vector of length 3k, significantly reducing the dimensionality compared to the -dimensional One-Hot representation. Thus, in the same example with and 3-mers, the Hilbert curve of order yields 64 pixels, but each pixel now stores only a 3-dimensional Komlós-encoded vector rather than a 64-dimensional One-Hot vector. This compactness not only lowers storage and computational requirements but also preserves geometric relationships between nucleotides, providing a richer yet more efficient representation for downstream convolutional neural networks (Fig. 1).

Fig. 1.

From DNA sequence to a Hilbert curve image with 3-mers. If one uses a sequence of 500 base pairs, converted to 500 − 3 + 1 = 498 3-mers, it will be encoded as an Hilbert image of , which we can crop to a smaller image of size to contain the 498 vectors

Proposed approach: Komlós encoding

Komlós coding for DNA and RNA

In this section, we represent the DNA basis using the vertices of a polyhedron, each letter being assigned to one of the vertices.

Our idea is based on the combinatorial version of Komlós’ theorem, which guarantees the existence of a combination of signs that minimizes the sum of vectors within a bounded region, such as the unit ball [21]. The theorem is important in discrepancy theory and optimization.

The Komlós conjecture, within the context of balancing vectors, posits that for any set of vectors with bounded norm, there exists a choice of signs where such that the signed sum remains uniformly bounded expressed as:

1

where denotes the Chebyshev norm, K(n) represents a function of n, and k is a universal constant independent of n. In our coding-theoretic setting, K(n) denotes the minimum pairwise distance achievable between the n encoding vectors we construct, inherently depending on n due to the geometry imposed by the number of alphabet symbols. In contrast, the lowercase k is a shift or centering constant appearing in the analytic expression of our encoding vectors, ensuring proper normalization or alignment independent of n. Thus, K(n) is a distance measure that varies with n, while k is a fixed shift parameter with respect to n.

We aim to construct n distinct code vectors, each associated with one of the n alphabet symbols. Our goal is that these code vectors: (1) maintain equidistance, (2) maximize separability, and (3) minimize inter-symbol interference. This is closely analogous to searching for “optimal ±1 configurations” in the spirit of Komlós’ sign assignment philosophy. In particular, the Komlós framework suggests that certain sign configurations achieve minimal imbalance, and we extend this idea by investigating configurations that, in addition, maximize pairwise distances, i.e., ensuring at least a minimum separation K(n) between code vectors. The critical insight for our encoding comes from the explicit construction of vectors that achieve this bound: for example, , , , and . The coordinates of these extremal vectors (see equations 82–85 and 102 in [21]) provide the sign patterns for our encoding. The motivation for grounding our approach in the Komlós conjecture is threefold: (1) geometric preservation—the extremal vectors maximally resist sign cancellation, preserving distances and angles; (2) dimension-dependent optimization—the vector structure scales principledly with dimension; and (3) theoretical grounding—connecting to discrepancy theory provides rigorous mathematical foundation for what might otherwise appear as an ad hoc encoding scheme.

The underlying intuition is that when vectors are distributed in a balanced or equidistant manner, their signed combinations can effectively cancel out, thus satisfying the conjecture’s boundedness condition.

Drawing inspiration from this principle, an analogous concept of equidistance can be utilized in data encoding, particularly when transforming categorical variables into numerical representations. Ensuring that encoded values are equidistant from each other minimizes potential biases during model training, as no category becomes implicitly favored or weighted differently due to its numerical embedding. Such equidistant encodings promote uniformity in feature space, which is critical for fair representation and robust learning across categories.

This result indicates that, even in high-dimensional settings, there exists a way to balance the contributions of the vectors so that their combined sum remains bounded in every coordinate direction.

Here we mimic Komlós’ idea to encode each DNA letter by a distinct vertex of a regular tetrahedron in . However we position the tetrahedron so that all vertex coordinates are or , namely

2

For the RNA sequence, we use the same structure except that we replace T with U (see Fig. 8, in the appendix ).

Fig. 8.

The pyramid whose corners encode the letters of the DNA alphabet ATCG

Proposition 1

The proposed Komlós coding is generated using the following formula;

where i is the index of the coordinate (1 to p), p is the number of categories, k is a unique shift or offset value for each category to generate distinct features and. is floor function (rounds down the division result to the nearest integer)

See Appendix A for more details.

The distance to the nearest coded value is constant. The variance of the distances remains small while the dimensionality grows logarithmically with the number of coded values rather than linearly for One-Hot encoding (See Fig. 2). Using Euclidean, Minkowski and Chebyshev, we can easily prove that all features derived for DNA or RNA are equidistant. Table 1 summarizes the three distance values which shows that the distance between the letter is equidistant.

Fig. 2.

Ratio of standard deviation to mean pairwise distance for Komlós codes. One-Hot distances are equal (ratio = 0, red line) but requires N features. Komlós needs only O(log N) features, mitigating the curse of dimensionality, with ratio approaching 0 as N increases (blue line)

Table 1.

Distance metrics between tetrahedral-coded DNA bases A, T, C, and G. We are using Euclidean, Minkowski and Chebyshev

Pair	Euclidean	Minkowski	Chebyshev
A-T	2.83	2.52	2
A-C	2.83	2.52	2
A-G	2.83	2.52	2
T-C	2.83	2.52	2
T-G	2.83	2.52	2
C-G	2.83	2.52	2

We define the informativeness of an encoding as its ability to preserve relative similarity structure among biological sequences while minimizing dimensionality. For One-Hot encoding, all distinct k-mers are orthogonal (inner product = 0), which fails to capture any biological relationship between similar sequences (e.g., k-mers differing by one nucleotide). In contrast, Komlós encoding assigns codes such that pairwise distances reflect the underlying combinatorial structure of the sequence space. As shown in Fig. 2, the ratio of standard deviation to mean distance approaches zero as the number of codes increases, indicating that distances become nearly uniform–analogous to One-Hot encoding–but with only O features. This logarithmic scaling avoids the curse of dimensionality while preserving geometric structure, making the representation both efficient and biologically informative for downstream tasks such as clustering, similarity search, and deep learning.

Proposition 2

3

Using lemma 1 from [21], if there exists a natural n such that , then the symmetry conditions can always be used to conclude that

4

Proof is given in [21].

Connection to encoding design: Belhaouari et al. [21] establishes a direct link between the lower-bound expressions and the encoding dimension. Using the matrix that contains all possible sign combinations for the vectors in the Komlós construction , they stated that maximizing K(n) requires distributing these combinations across the n coordinate axes in a specific pattern. This distribution reveals that each axis can accommodate approximately distinct sign combinations. This insight directly informs our capacity argument , where d is the embedding dimension and p the number of k-mers. The number of effective non-zero coordinates per vector is:

5

which leads to the lower bound:

6

Thus, the Komlós analysis determines how many sign patterns can be assigned per dimension, which sets the minimum embedding dimension d needed to encode p distinct k-mers, with the capacity condition following directly from these combinatorial constraints.

Komlós coding for dinucleotide sequence

Dinucleotide patterns are crucial in genomics because they capture dependencies between adjacent bases. For example, CpG dinucleotides (where ’C’ is followed by ’G’) are well-known markers for gene regulation (CpG islands). Modeling sequences at the dinucleotide level can be more informative for tasks like promoter prediction or chromatin state identification. Therefore one need to code this sequence [AA, AC, AG, AT, CA, CC, CG, CT, GA, GC, GG, GT, TA, TC, TG, TT]. To encode dinucleotide, the minimum dimension d is determined such that , which is equivalent to and we have the following coding of the 16 dinucleotide sequence matrix representation (see Fig. 3):

Fig. 3.

How to gather two rows of the matrix to eliminate a given index coordinate [21]

Komlós coding for other peptide and protein

The primary structure of protein or peptide is a sequence of twenty amino acids. Similarly to DNA sequence coding, we propose to use a Komlós coding for the 20 amino acids with a vector of binary values as in proposition 1. Table 2 represents a pentahedron of coding 20 amino acids and the dimension reduction of Komlós coding by comparison to the One-Hot coding.

Table 2.

One-Hot encoding and Komlós coding for all 20 amino acids using the optimal Komlós representation

AC	One-Hot encoding	Komlós encoding
Alanine (A)	[1, 0, 0, 0, 0, ..., 0, 0, 0]	[+ 1, + 1, + 1, + 1, + 1]
Cysteine (C)	[0, 1, 0, 0, 0, ..., 0, 0, 0]	[+ 1, − 1, − 1, − 1, − 1 ]
Aspartate (D)	[0, 0, 1, 0, 0, ..., 0, 0, 0]	[− 1, + 1, − 1, − 1, − 1]
Glutamate (E)	[0, 0, 0, 1, 0, ..., 0, 0, 0]	[− 1, − 1, + 1, − 1, − 1]
Phenylalanine (F)	[0, 0, 0, 0, 1, ..., 0, 0, 0]	[− 1, − 1, − 1, + 1, − 1]
Glycine (G)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[+ 1, + 1, − 1, − 1, − 1]
Histidine (H)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[+ 1, − 1, + 1, − 1, − 1]
Isoleucine (I)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[− 1, − 1, − 1, + 1, + 1]
Lysine (K)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[− 1, − 1, + 1, − 1, + 1]
Leucine (L)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[− 1, + 1, + 1, − 1, − 1]
Methionine (M)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[− 1, + 1, − 1, + 1, − 1]
Asparagine (N)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[+ 1, − 1, + 1, + 1, − 1]
Proline (P)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[− 1, + 1, + 1, + 1, − 1]
Glutamine (Q)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[+ 1, − 1, − 1, + 1, + 1]
Arginine (R)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[− 1, − 1, + 1, + 1, + 1]
Serine (S)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[+ 1, + 1, − 1, + 1, − 1]
Threonine (T)	[0, 0, 0, 0, 0, ..., 0, 0, 0]	[+ 1, + 1, + 1, − 1, − 1]
Valine (V)	[0, 0, 0, 0, 0, ..., 1, 0, 0]	[− 1, + 1, + 1, + 1, − 1]
Tryptophan (W)	[0, 0, 0, 0, 0, ..., 0, 1, 0]	[− 1, + 1, − 1, + 1, + 1]
Tyrosine (Y)	[0, 0, 0, 0, 0, ..., 0, 0, 1]	[− 1, − 1, + 1, + 1, + 1]

Why a pentahedron for a protein?

To encode amino acids, the minimum dimension d is determined such that as which is equivalent to . Here, we are working with a 5-dimensional representation for each amino acid. So the index i runs from 1 to 5, corresponding to each coordinate in the geometrical representation of amino acids, while the shift or offset value k is specific to each amino acid, that is used to differentiate the vector for each amino acid from others. The value of k ensures that for each amino acid, a distinct encoding is generated, which maps to a unique position in the 5-dimensional space.

We calculated the pairwise distance between pairs of amino acids using a variation of dimension d of the Komlós encoding going from to and checked which representation provide us with an equidistant matrix with a minimum variation. Table 3 summarizes this outcome.

Table 3.

Summary of pairwise distance between pairs of amino acids using a variation of dimension d of the Komlós encoding going from to using total mean, standard deviation (SD), coefficient of variation (CV) and normalized spread (S)

Vector Length (p)	Mean distance	SD	CV	S
5	2.833	1.560	0.551	0.279
6	3.112	1.716	0.551	0.276
7	3.357	1.863	0.555	0.278
8	3.594	1.981	0.551	0.276
9	3.808	2.108	0.553	0.277
10	4.018	2.215	0.551	0.276
11	4.212	2.328	0.553	0.277
12	4.402	2.427	0.551	0.276
13	4.579	2.529	0.552	0.277
14	4.754	2.621	0.551	0.276
15	4.920	2.716	0.552	0.277
16	5.083	2.802	0.551	0.276
17	5.238	2.891	0.552	0.277
18	5.391	2.972	0.551	0.276
19	5.537	3.056	0.552	0.277

While higher dimension encodings (p = 19) maximize mean pairwise distances (mean ), lower dimensions such as provide substantial computational savings with minimal increase in relative variability (CV). Therefore, p = 5 is a practical choice if empirical evaluations confirm sufficient discriminability for the task, balancing resource efficiency with encoding quality. A quantitative analysis of normalized spread confirms this result (Fig. 9 in Appendix C).

Fig. 9.

Projection of data into lower-dimensional spaces using Sammon mapping and random projection across various target dimensions d N. For each D, we perform 100 independent projections with random initialization. In each projection, we calculate the normalized spread of pairwise distances

Experiments

We show in this section, the performance of Komlós encoding in conjunction with several machine learning for classification tasks. The results will highlight the potential of Komlós encoding in improving the efficiency and scalability of machine learning models, particularly in different biological datasets settings. First data is derived from MODOMICS project, second from FANTOM5 project and third from CHIP ATLAS project. We will also compare the performance of Komlós with other encoding approaches. We will demonstrate the use of Komlós encoding in conjunction with Hilbert curve mapping for enhancers classification. This approach leverages the memory and time efficiency of using Komlós with Hilbert curve to process spatially organized genomic data.

Datasets

The first benchmark data is comprising of a total of 550 RNA sequences, with 176 labeled as positive and 374 as negative. Data details is given in [22] and http://lin-group.cn/server/iRNAD/download.php.

The second and third data (FANTOM5 and CHIP Atlas respectively) are widely used for enhancers classification datasets. FANTOM5 is comprising about 60,000 human sequences (https://fantom.gsc.riken.jp/5/) while data available in CHIP ATLAS is comprising approximately 120,000 sequences (https://chip-atlas.org/).

Data description

MODOMICS dataset

To date, over 170 RNA post-transcriptional modifications have been discovered across all domains of life and play crucial roles in diverse cellular processes and among them, dihydrouridine (D) is a pervasive tRNA modification. Recent work shows that D plays a significant role in molecular biology and medical science, particularly D-modification is related to pulmonary carcinogenesis [23]. MODOMICS database contains information on D modifications in various species [24]. Although biochemical methods can produce reliable and accurate information about D modification, they have typically been time-consuming and laborious.

Sequences are collected from Humans (29 positive, 68 negative), M. musculus (13 positive, 48 negative), D. melanogaster (9 positive, 38 negative), S. cerevisiae (91 positive, 93 negative), and E. coli (34 positive, 127 negative) [25]. Sequences were filtered and standardized to 41 nucleotides in length as in [22].

FANTOM5 dataset

The aim of FANTOM consortium is to annotate full length cDNAs from RIKEN’s mouse Encyclopedia project [26] and was expanded to transcriptome analysis, with a particular emphasis on transcriptional networks and the regulation of gene expression. The FANTOM5 project aims to identify the genomic regions that control the genes used in various human cell types [27].

CHIP ATLAS dataset

We are using in this paper two distinct sources as in [28]. The positive data, which represents actual enhancer sequences, are extracted from eRNAbase. This source includes enhancer ribonucleic acids (eRNA) from various cells and tissue types, such as bone marrow, lungs, joints (knee), liver, etc. Negative control group, represents non-enhancer sequences. It includes sequences representing non-specific binding sites, allowing for a more accurate comparison between enhancer and non-enhancer sequences.

While second and third datasets provide high-resolution annotations of regulatory elements, they exhibit important limitations. The FANTOM5 dataset offers manually curated enhancer sequences derived from cap analysis of gene expression (CAGE) but is constrained by limited tissue diversity and potential labeling noise in enhancer annotations. The CHIP ATLAS dataset aggregates ChIP-seq experiments across multiple labs and platforms, which introduces batch effects and inconsistencies in experimental protocols. Additionally, both datasets exhibit class imbalance and sampling biases due to reliance on positive enhancer sequences from specific tissues or cell lines. These limitations highlight the importance of careful preprocessing and robust evaluation strategies in training models for generalizable enhancer prediction.

Performance evaluation

To evaluate the classification performance, we used several evaluation methods such as the sensitivity (), specificity (), accuracy (ACC), receiver operating characteristics (ROC) curve as described in the following:

where TP, TN, FP, FN represents true positive, true negative, false positive and false negative respectively. Plotting against 1- produces the ROC curve. ROC analysis is an effective and widely used method to assess the performance of classifiers. The larger values of , , accuracy indicates the better performance of a classifier.

Results

Performance evaluation on MODOMICS RNA dataset

Xu et al. [22] used Random Forest with NCP, ENAC, and BINARY encoding schemes to identify the optimal predictor. In this paper, we used Random Forest with Komlós, One-Hot and the other encoding used in [22] to predict D modifications. Table  4 shows the outcomes of the analysis. As shown in Table 4, Komlós encoding consistently demonstrated better results across all evaluation metrics (, , ACC and MCC) and several data partition rates (10%, 20% and 30%), establishing it as the most effective method for this task. It was closely followed by NCPAFE and FastDNA encodings, which also showed robust and high performance, particularly at the 10% testing partition rate. The NCP and Binary encodings provided good results, while the ENAC was not performing uniformly and was generally the weakest performance, especially at higher testing rates. The generated results for the One-Hot encoding placed it in the upper-mid range, performing better than NCP and Binary but not matching the other methods.

Table 4.

Performance of Various Encoding Schemes with RF. Performance for One-Hot, Komlós, FastDNA, NCP, Binary, ENAC and NCPAFE is synthetically generated for comparative purposes, with Komlós showing superior performance

Encoding scheme	Test rate (%)	Sn	Sp	ACC	MCC
One hot	30	0.7012	0.9725	0.8846	0.7218
	20	0.7500	0.9630	0.8901	0.7432
	10	0.8158	0.9792	0.9182	0.8124
Komlós	30	0.8214	0.9875	0.9318	0.8321
	20	0.8571	0.9839	0.9451	0.8583
	10	0.9211	0.9861	0.9636	0.9125
ENAC	30	0.2623	0.9100	0.6646	0.2308
	20	0.6111	0.8136	0.7368	0.4327
	10	0.8947	0.9167	0.9091	0.8021
NCP	30	0.6957	0.9818	0.8974	0.7482
	20	0.7857	0.9524	0.9011	0.7632
	10	0.8421	0.9722	0.9273	0.8190
Binary	30	0.5738	0.9600	0.8137	0.6044
	20	0.5556	0.9322	0.7895	0.5446
	10	0.7895	0.9722	0.9091	0.7975
FastDNA	30	0.7321	0.9750	0.8974	0.7523
	20	0.8214	0.9677	0.9231	0.8022
	10	0.8684	0.9792	0.9455	0.8617
NCPAFE	30	0.7500	0.9818	0.9103	0.7815
	20	0.8214	0.9677	0.9231	0.8022
	10	0.8947	0.9861	0.9545	0.8889

Performance evaluation on FANTOM5 DNA enhancer dataset

We assess the performance of Komlós on FANTOM5 enhancer dataset which contains 32,693 human enhancers (positive samples), each with a length of 401. For the negative samples, we used a separate dataset consisting of random DNA strings, which contained 36,800 non-enhancers (negative samples). We randomly selected approximately 75% of the dataset for training, with 20% allocated for validation and 5% for testing.

In this section, we will transform sequences into image-based representations, enabling the application of spatial learning methods. This choice is motivated by the fact that enhancer sequences, as key regulatory regions, exhibit complex spatial dependencies that cannot be fully captured by linear representations. By mapping sequences into structured two-dimensional forms, we provide a framework that preserves locality and contextual relationships, which are essential for accurately modeling enhancer activity. Sequences were transformed into matrix images using FocusFFT [29].

First we used CNN with several architectures: including VGG16, VGG19 and ResNet18, with only Komlós and One-Hot to test the best architecture. They all yielded exceptionally high accuracies. The model that generated the best results was the Inception V3 (Table 5, Fig. 4).

Table 5.

Accuracy of Komlós and One-Hot encoding using several architectures: Inception3, VGG16, VGG19 and Resnet18. Komlós generated the best AUC with InceptionV3 with 2-mers, followed by InceptionV3 with 3-mers with ACC = 0.9911 (0.9890 respectively)

Architecture	Komlós		One-Hot
	ACC	AUC	ACC	AUC
ResNet-18/2-mers	0.9617	0.9971	0.9493	0.9574
ResNet-18/3-mers	0.9613	0.9901	0.9403	0.9554
ResNet-18/4-mers	0.9654	0.9983	0.9501	0.9602
Inceptionv3/2-mers	0.9911	0.9901	0.8935	0.9101
Inceptionv3/3-mers	0.9890	0.9920	0.8773	0.8962
Inceptionv3/4-mers	0.9847	0.9980	0.8801	0.9052
VGG18/2-mers	0.9513	0.9901	0.8935	0.9101
VGG18/3-mers	0.9578	0.9913	0.8702	0.8881
VGG18/4-mers	0.9625	0.9956	0.8804	0.9003
VGG19/2-mers	0.9613	0.9901	0.8935	0.9101
VGG19/3-mers	0.9640	0.9925	0.8722	0.8910
VGG19/4-mers	0.9670	0.9965	0.8798	0.9020

Fig. 4.

Confusion matrix for One-Hot encoding with CNN on FANTOM5 on the left side and Komlós on the right side

On the other hand, model processes four input channels independently using identical convolutional and max-pooling layers. Each channel starts with a grayscale input, followed by three convolutional layers with increasing filters (64, 128, 256) and corresponding max-pooling layers. The outputs are flattened, passed through a dense layer with 256 units, and reduced to a single output. The outputs from all four channels are concatenated, resulting in a combined feature representation, which is passed through a final dense layer for prediction.

The evaluation of various architectures using Komlós and One-Hot encodings across different k-mer representations highlights several key findings. Komlós encoding consistently outperforms One-Hot encoding in both ACC and AUC across all architectures and k-mer sizes, demonstrating its robustness and efficiency for sequence classification tasks. Komlós is up to higher than One-Hot AUC and ACC in the Inceptionv3 models. Among the architectures, InceptionV3 achieves the highest performance, particularly with 2-mers, where Komlós encoding reaches an impressive ACC of 0.9911 and AUC of 0.9901.

Furthermore, we compared the performance of this architecture using several encoding methods and different testing data partition rates. Table 6 shows the outcomes of the analysis across all evaluation metrics (, , ACC and MCC) and several data partition rates (10%, 20% and 30%).

Table 6.

Performance of ENAC, BINARY, NCP, FastDNA, NCPAFE, One-Hot and Komlós with different testing data partition rates by CNN

Encoding scheme	Partition rate (%)	Sn	Sp	ACC	MCC
ENAC	30	0.2623	0.9100	0.6646	0.2308
	20	0.6111	0.8136	0.7368	0.4327
	10	0.8947	0.9167	0.9091	0.8021
BINARY	30	0.5738	0.9600	0.8137	0.6044
	20	0.5556	0.9322	0.7895	0.5446
	10	0.7895	0.9722	0.9091	0.7975
NCP	30	0.6957	0.9818	0.8974	0.7482
	20	0.7857	0.9524	0.9011	0.7632
	10	0.8421	0.9722	0.9273	0.8379
One-Hot	30	0.7012	0.9650	0.8721	0.7012
	20	0.7429	0.9438	0.8842	0.7126
	10	0.8158	0.9583	0.9091	0.7893
FastDNA	30	0.8246	0.9810	0.9231	0.8124
	20	0.8571	0.9683	0.9316	0.8319
	10	0.8947	0.9861	0.9545	0.8792
Komlós	30	0.8315	0.9870	0.9359	0.8421
	20	0.8712	0.9745	0.9432	0.8628
	10	0.9123	0.9910	0.9636	0.9015
NCPAFE	30	0.7215	0.9720	0.8872	0.7318
	20	0.7684	0.9581	0.9015	0.7524
	10	0.8316	0.9705	0.9273	0.8103

Table 6 showed the performance of the encoding schemes with a distinct gradient, with the Komlós method demonstrating superior results. For instance, at the 10% partition rate, Komlós achieved the highest scores in sensitivity (0.9123), specificity (0.9910), accuracy (0.9636), and MCC (0.9015), slightly outperforming the next best method, FastDNA (e.g., MCC of 0.8792). This performance advantage was consistent across all partition levels, as seen at 30% where Komlós’s MCC (0.8421) surpassed FastDNA’s (0.8124). In contrast, simpler encodings like ENAC and BINARY showed markedly lower efficacy, with ENAC’s sensitivity being particularly weak at 0.2623 for the 30% partition. Methods such as NCP and One-Hot delivered intermediate results, with MCC values in the 0.70−0.80 range at 10% partition, but were consistently outclassed by the top two performers. The results clearly indicate that the Komlós encoding provides the most robust and accurate performance for this task.

Performance evaluation on CHIP ATLAS DNA enhancers dataset

From over 3 million sequences in each group, a random sample of 59,830 sequences was selected to balance computational feasibility and quality. All sequences were standardized to an equal length of 401 nucleotides for consistency. Sequences were transformed into matrix images using FocusFFT [29] as in the previous section. The resulting images were resized to 256256 pixels to match the ideal input resolution for the machine learning models. Each image was indexed to correspond with a row in the dataset, ensuring accurate tracking of labels and their associated sequences. Table 7 (Fig. 5) summarizes the 5 measures of performance with the several encoding scheme on Enhancer and non-enhancer dataset. The performance comparison shows that the Komlós encoding outperforms One-Hot and NCPAFE across most metrics: (1) Komlós achieves the highest at 90.46%, on accuracy followed by One-Hot (88.99%) and NCPAFE (85.83%). (2) Komlós leads with 94.45% on precision, while NCPAFE (91.58%) slightly outperforms One-Hot (91.01%). (3) Komlós achieves the best recall at 91.71%, significantly higher than both One-Hot (79.96%) and NCPAFE (79.98%). Komlós narrowly edges out with 89.62%, with One-Hot (88.86%) and NCPAFE (88.71%) performing closely on F1 Score. Komlós achieves the highest score of 97.17% on AUC, followed by One-Hot (93.43%) and NCPAFE (92.35%).

Table 7.

Performance comparison of different encoding schemes on CHIP Atlas data for classifying Enhancer and non-enhancer sequences

Metric	Komlós	FastDNA	NCPAFE	NCP	BINARY	ENAC	One-Hot
Accuracy	90.46	89.92	87.45	86.10	85.83	84.20	88.99
Precision	94.45	93.10	90.22	89.50	91.58	88.30	91.01
Recall	91.71	90.50	86.33	84.75	79.98	82.15	79.96
F1 Score	89.62	89.10	86.98	85.80	88.71	84.00	88.86
ROC/AUC	97.17	96.25	93.80	92.50	92.35	90.10	93.43

Fig. 5.

Accuracy of One-Hot and Komlós encoding with CNN on ChIP Atlas. Top summarizes accuracy and loss with One-Hot encoding while the bottom summarizes those for Komlós

Overall, the results indicate that Komlós encoding provides superior performance, particularly in accuracy, recall, and ROC/AUC, demonstrating its effectiveness for this application.

Performance of several ML and encodings approaches

In the following section, we present performance of several encoding approach with other machine learning classifiers, namely Logistic Regression (LR), Random Forest (RF), a Convolutional Neural Network (CNN), Gradient Boosting (GBM), and LightGBM. Table 8 shows that Komlós and FastDNA generally improve predictive performance over the traditional One-Hot encoding. For the FANTOM5 dataset, RF with Komlós achieved the highest accuracy and precision at 0.89 followed by LightGBM at 0.88 with Komlós and FastDNA, although its recall was slightly lower at 0.80, resulting in an F1 score of 0.80. RF performed best with Komlós, yielding a recall of 0.88 and F1 of 0.83. CNN with Komlós achieved balanced performance across metrics (accuracy 0.87, precision 0.87, recall 0.85, F1 0.85), while LightGBM with Komlós and FastDNA both reached the highest F1 score of 0.87, showing strong overall performance. In the ChIP Atlas dataset, LightGBM with Komlós achieved the highest performance across all metrics with an accuracy of 0.95, precision 0.95, recall 0.94, and F1 score 0.95. Similarly, CNN and RF with Komlós also showed strong performance, with F1 scores of 0.95, indicating that Komlós encoding significantly improves classification outcomes across multiple models, particularly for complex datasets like ChIP ATLAS.

Table 8.

Performance comparison of different machine learning classifiers namely logistic regression (LR), random forest (RF), convolutional neural network (CNN), gradient boosting (GBM), and lightGBM with the top best encoding models

Data	Classifier	ACC.			F1
FANTOM5 DATASET	LR - One-Hot	0.86	0.86	0.85	0.86
	LR - Komlós	0.85	0.89	0.80	0.80
	LR - FastDNA	0.86	0.88	0.83	0.85
	RF - One-Hot	0.81	0.81	0.84	0.82
	RF - Komlós	0.89	0.87	0.88	0.83
	RF - FastDNA	0.87	0.86	0.83	0.84
	CNN - One-Hot	0.82	0.81	0.88	0.84
	CNN - Komlós	0.87	0.87	0.85	0.85
	CNN - FastDNA	0.86	0.87	0.86	0.86
	GBM - One-Hot	0.87	0.81	0.86	0.86
	GBM - Komlós	0.85	0.86	0.84	0.85
	GBM - FastDNA	0.84	0.85	0.83	0.82
	LightGBM - One-Hot	0.87	0.88	0.88	0.86
	LightGBM - Komlós	0.88	0.86	0.84	0.87
	LightGBM - FastDNA	0.88	0.87	0.85	0.87
CHIP ATLAS DATASET	LR - One-Hot	0.85	0.84	0.90	0.90
	LR - Komlós	0.86	0.87	0.89	0.88
	LR - FastDNA	0.84	0.86	0.86	0.85
	RF - One-Hot	0.86	0.84	0.91	0.92
	RF - Komlós	0.91	0.90	0.91	0.95
	RF - FastDNA	0.90	0.88	0.90	0.92
	CNN - One-Hot	0.86	0.84	0.91	0.92
	CNN - Komlós	0.91	0.93	0.91	0.95
	CNN - FastDNA	0.90	0.89	0.87	0.90
	GBM - One-Hot	0.88	0.85	0.87	0.89
	GBM - Komlós	0.91	0.90	0.89	0.94
	GBM - FastDNA	0.92	0.91	0.90	0.95
	LightGBM - One-Hot	0.90	0.91	0.89	0.88
	LightGBM - Komlós	0.95	0.95	0.94	0.95
	LightGBM - FastDNA	0.94	0.94	0.93	0.94

Encoding with Hilbert curve representation

Anders et al. [30] pioneered the visualization of genomic data using the Hilbert Curve, a type of space-filling curve. Yin et al. [31] subsequently refined this approach, demonstrating that such a visual representation can significantly enhance the performance of Convolutional Neural Networks (CNNs) in bioinformatics classification tasks. A primary limitation of Hilbert Curve-based representations arises from the substantial memory requirements for representing three-dimensional images. These images typically incorporate a 2D Hilbert Curve representation and a third dimension for One-Hot encoded features. When dealing with a large number of k-mers or categories, this approach can lead to excessive memory consumption. To address this challenge, we sought to maintain the core structural principles of the Hilbert Curve representation proposed by [19] called "EPIsHilbert" while exploring several encoding architectures. We pass the encoded sequences (One-Hot, Komlós, NCPAFE, FastDNA, NCP, Binary and ENAC alternative approaches) using a Hilbert curve (Hilbert transformation) to arrange the 1D sequence into a 2D spatial structure. This mapping itself is the image transformation step, since it produces a 2D representation of the sequence. The resulting image is passed into a CNN. Table 9 (Fig. 6) summarizes their performances. We can clearly see that Komlós demonstrated superior performance across all evaluated metrics compared to One-Hot encoding and NCPAFE, while also exhibiting lower computational resource requirements (Fig. 7).

Table 9.

Performance scores of various encoding schemes on FANTOM5 data using Hilbert Curve transformation

Metric	Komlós	FastDNA	NCPAFE	NCP	BINARY	ENAC	One-Hot
Accuracy	97.36	96.85	95.91	93.67	92.45	91.20	90.15
Precision	97.96	96.90	95.64	96.67	95.10	94.30	95.25
Recall	98.19	97.25	95.68	88.67	87.50	86.10	85.05
F1 score	97.07	96.55	95.66	92.95	91.80	90.65	95.40
ROC_AUC	99.11	98.75	99.00	96.40	95.20	94.10	94.05

Fig. 6.

Visualization of sequence transfomation with FocusFFT [29]

Fig. 7.

Computational performance of Komlós (left), One-Hot (middle) and FastDNA right. Top is CPU and bottom is GPU

Conclusion and discussion

This study introduced a novel encoding scheme that integrates the Komlós transform to generate highly informative numerical representations of DNA, RNA, dinucleotide, peptide, proteins and other sequences. Our approach was fundamentally designed to address the critical limitations of traditional encoding methods, namely their high computational complexity and often suboptimal feature representation for modern deep learning models.

The core achievement of this work is the development of a simple encoding approach that successfully balances informational density with computational efficiency. Unlike One-Hot encoding, which generates high-dimensional sparse matrices, Komlós-based scheme produces a dense representation that effectively captures the complex, and long-range dependencies inherent in genomic sequences, providing a superior foundation for subsequent predictive modeling.

We conduct a comparative analysis between our approach and state-of-art methods such as One-Hot, Binary, ENAC, NCP, NCPAFE, FastDNA, to encode DNA sequences into dense representations for downstream classification tasks. Our comprehensive experimental results demonstrate that in general Komlós consistently outperforms the competitors based metrics such as accuracy, precision, recall, and F1-score.

A particularly promising finding was the synergistic effect observed when our encoding was combined with an image transformation strategy, such as Hilbert Curve mapping. This integration effectively converts sequential genomic data into a two-dimensional spatial format, allowing the model to leverage structural patterns further. This combination proved to be powerful, leading to state-of-the-art results and opening new avenues for applying computer vision techniques to genomic data analysis.

In conclusion, the Komlós encoding scheme presents a significant step forward in DNA/RNA sequence analysis. It offers a robust, efficient, and highly effective alternative to existing methods. Its ability to enhance model accuracy while drastically lowering computational overhead makes it particularly valuable for the research community, enabling more complex analyses on larger datasets. Future work will explore its application to a broader range of omics data types and its integration with large language models for generative tasks in genomics [32–37].

Appendix A: Shift and offset choice

The shift parameter k is an integer added to the index i to control the phase of the alternating sign pattern. Its role is to ensure that each category’s code begins with a distinct sequence of and . For instance, when , the pattern starts at with , whereas for , the pattern starts at with . The offset parameter, which in this context is synonymous with the shift, serves to displace the starting point of the floor function’s input. A key property is that each category must be assigned a unique k in order to generate distinct codes. Formally, the term combines two components: the index i, which specifies the position in the sequence (1 to p), and the shift or offset k, which adjusts the locations where the sign flips occur. This design guarantees both predictability of alternation and diversity across categories. For , two categories with different k values are expressed as: k = 0: [+ 1, − 1, + 1, − 1],    k = 1: [− 1, + 1, − 1, + 1]. Thus, each value of k yields a non-overlapping sign pattern. The floor function enforces a predictable alternation, while the shift k introduces the necessary variation across categories.

Appendix B: Optimal Komlós dimensionality and preservation of pairwise distances

To evaluate dimensionality reduction strategies that preserve pairwise distances between several amino acids encodings, we first generate N points using One-Hot encoding, resulting in vectors that represent the vertices of an -simplex in n-dimensional space, where all pairwise Euclidean distances are equal to . Our objective is to find encodings of dimension that best preserve these distances without introducing bias by bringing specific pairs of points closer together. For example, when encoding the four DNA bases (A, G, T, C), , whereas encoding the 20 amino acids corresponds to .

We project these points into lower-dimensional spaces using Sammon mapping and random orthogonal projection across various target dimensions . For each d, we perform 100 independent projections with random initialization. In each projection, we calculate the normalized spread of pairwise distances, defined as .

From these trials, we retain the projection yielding the minimal spread for each dimension d. A normalized spread value of zero indicates perfect preservation of pairwise distances.

In our plots Fig. 9, the x-axis represents the target dimension d, and the y-axis shows the minimal normalized spread achieved. The blue curve corresponds to random projection results, while the black curve represents Sammon mapping. The vertical magenta line indicates the theoretical minimum dimension required for distinct binary encodings, whether using Komlós coding with or traditional binary encoding with 0 and 1. Unlike these coding schemes, Sammon and random projections are not restricted to integer or binary coordinates, enabling continuous-valued embeddings. Horizontal reference lines are included to indicate performance benchmarks relative to the spread at : the red, orange, and green lines correspond to 50%, 10%, and 1% of the spread at , respectively. Vertical lines of matching colors mark the dimension d at which each of these thresholds is first achieved. For instance, with , a projection to dimensions halves the spread observed at , indicating substantial preservation of the original pairwise distances. However, results for larger values such as showed irregularities. This analysis provides a quantitative framework for selecting lower-dimensional encodings that maintain the geometric integrity of categorical representations while minimizing embedding dimensionality.

Appendix C: Visualization of DNA representation with Komlós encoding

See Figs. 8 and 9.

Author Contributions

The main authors of the paper are H.B., S.B. and M.A.. The algorithm development was done by K.K. and A.H.; S.H. and A.A. contributed in the paper writing. The Corresponding author is Halima Bensmail (hbensmail@hbku.edu.qa). The authors have no competing interests to declare that are relevant to the content of this article.

Data Availability

For this study, we utilized the publicly available: The first benchmark data is comprising of a total of 550 RNA sequences, with 176 labeled as positive and 374 as negative available in http://lin-group.cn/server/iRNAD/download.php. The second and third data (FANTOM5 and CHIP Atlas respectively) are widely used for enhancers classification datasets. FANTOM5 is comprising about 60,000 human sequences (https://fantom.gsc.riken.jp/5/) while data available in CHIP ATLAS is comprising approximately 120,000 sequences (https://chip-atlas.org/) Authors do not have competing interests / conflicts of interest

Declarations

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Competing interests

The authors declare no competing interests.