## 1 Introduction
First, any signature image is embedded to the feature space under an appropriate coding scheme; this is formally termed the signature descriptor. In the second step, sets of (dis)similar descriptors emerging from signature pairs conditioned under the \(\omega^{\pm}_{WI}\) classes are employed as inputs to the learning stage of the verifier model in order to accurately determine its optimal operating parameters. Finally, the resulting verifier model is tested against pairs of signatures that have never contributed to the learning stage in order to determine whether they came from the same author or not. WI is more challenging for accuracy; early attempts were not as effective as their WD counterparts. Therefore, WI systems have been the primary subject of recent works [18,23,30-37]. Although SV is generally acknowledged as a binary classification task, the discrimination between \(\omega^{+},\ \omega^{-}\) is affected by the way that the negative class \(\omega^{-}\) is represented. In the context of SV [38], at least two different kinds of forgeries are identified: the first is random forgery (RF) (or zero-effort forgeries or intrinsic failure) in which the negative class is represented by samples that originate from someone other than the authentic person (or an attacker). The second is skilled forgery (SF) or mimicry in terms of biometric presentation attacks. These samples incorporate a portion of knowledge of the signing process and usually exhibit greater resemblance to those of authentic users compared to random forgeries, which affects the accuracy of signature verifiers. For the WI case, negative class representatives can originate either from random forgeries or skilled forgeries. For the WI verifier, the positive or similar class, denoted by \(\omega^{+}\), is represented by genuine-to-genuine (G-G) pairs. The negative or dissimilar class, denoted by \(\omega^{-}\), can be represented either by genuine-to-random-forgery (G-RF) or genuine-to-skilled-forgery (G-SF) pairs or a mixture of them. Contrary to the restriction of the WD design stage, WI verifiers are allowed to utilize G-SF pairs only if the testing stage is blind or disjoint; that is, it tests pairs of signatures that do not contribute during the design-learning stage [28; 39; 40; 41; 42]. In this concept, an intuitively transfer learning approach by means of (a) a typical 5 \(\times\) 2 internal fold and/or (b) a cross-lingual design-testing external protocol [18; 19; 35; 43; 44; 45; 46] can be applied without the concern of inducing bias to our system. The WI-SV framework has been addressed under a number of \"handcrafted\", as well as contemporary, deep neural topologies. Perhaps the most common \"handcrafted\" WI-SV topology utilizes the Dichotomy Transform (DT), a technique that transforms a Polychotomizer--or in the context of SV, a writer-dependent approach--into a simple Dichotomizer, enabling a writer-independent approach. For the last fifteen years [21; 41], the DT has been applied for WI-SV purposes on a number of publications. Initially introduced by [PERSON] and [PERSON] [47] at the beginning of the millennium, the DT converts feature vectors to distance vectors by means of a simple mathematical operation. In the following years, the works of [PERSON] and [PERSON] [48; 49], and [PERSON] et al. [50], provided a theoretical foundation and justified the use of the dissimilarity space for recognition applications. Therefore, the DT is a legitimate candidate for designing WI-SV systems. The process of verifying an individual's signature is typically regarded as a visual recognition task. Up until now, any contemporary prevailing topologies, in addition to the previous utilization of the DT, were predicated on the underlying assumption that the data in the form of the signature descriptors form a vector space which complies with Euclidean space axioms [51]. But now, there is evidence that ignoring the geometrical properties of the data is restrictive [51].