Logarithmic Source Representation

• Logarithmic Source Representation is a collection of methods that employ logarithmic transforms to encode objects and reveal properties like positivity and concavity in diverse mathematical settings.
• It underpins analytic series, integral and probabilistic representations, and log-domain quantization in neural networks to improve both theoretical insights and computational efficiency.
• The approach extends to operator theory, inverse problems, PDEs, and geometric frameworks, thereby facilitating robust representations in contexts from unbounded generators to defect fields.

“Logarithmic Source Representation” is not a single standardized construction. Across the cited literature, it denotes several distinct but structurally related practices in which a logarithm, a logarithmic loss, or a logarithmic transform is used to encode the object of interest: a scalar function, a source reconstruction, a neural-network tensor, an infinitesimal generator, a nonlocal operator, or a defect-local field (Bradley, 2012, Courtade et al., 2011, Miyashita et al., 2016, Iwata, 2020, Chen et al., 2017, Shimamori et al., 15 Oct 2025). In each case, the logarithmic layer reorganizes the underlying problem: it can expose positivity and concavity, convert residual uncertainty into an entropy, replace multiplication by shifts or exponent addition, regularize unbounded generators through bounded logarithms, or encode localized disorder through Jordan-block defect operators.

1. Analytic representations of the logarithm

A foundational usage is the direct representation of the logarithm itself. Bradley derives a globally convergent infinite series

$$x-1-\log x=\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,\qquad x>0,$$

equivalently

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$

with convergence on the full natural domain $x>0$ rather than only on the local interval associated with the Taylor series for $\log(1+u)$ (Bradley, 2012). The derivation uses the sequence $a_k=2^k(x^{2^{-k}}-1)$, whose limit is $\log x$, and rewrites $x-1-\log x$ as a telescoping sum that becomes a sum of manifestly nonnegative squares. The same paper proves convergence by limit comparison with $\sum 2^{-k}$, obtaining the asymptotic term

$$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$

and uses termwise positivity to make the inequality $\log x<x-1$ and the concavity of $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$0 transparent. Through Jensen’s inequality, that concavity yields the arithmetic–geometric mean inequality (Bradley, 2012).

A second analytic line uses integral representations. One exact identity is

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$1

extended in the paper to $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$2 (Merhav et al., 2019). This converts expectations of logarithms into Laplace-transform expressions, such as

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$3

and similarly for $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$4 when the $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$5 are positive i.i.d. The paper applies this to entropy, differential entropy, empirical entropy, universal compression redundancy, and ergodic capacity, and presents the method as a rigorous alternative to replica-style manipulations in some settings (Merhav et al., 2019).

A third line concerns divergent Mellin-type integrals with logarithmic singularities. For finite-part integrals of the form

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$6

the logarithmic factor is generated by parameter differentiation,

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$7

and the finite part admits exact contour-integral representations (Ylanan et al., 19 Feb 2026). For noninteger $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$8, the coefficients are expressed through Stirling numbers of the second kind; for integer $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$9, Bernoulli numbers appear after taking a regularized limit. This extends earlier contour representations of Hadamard finite parts to arbitrary logarithmic order $x>0$0 (Ylanan et al., 19 Feb 2026).

2. Probabilistic source representation under logarithmic loss

In information theory, “logarithmic source representation” acquires a precise operational meaning: the decoder does not reproduce a source symbol by a hard decision, but by a probability distribution over source symbols (Courtade et al., 2011). For finite alphabet $x>0$1, the reproduction alphabet is

$x>0$2

and the per-letter distortion is

$x>0$3

Under this loss, reconstruction is best understood as posterior representation. If the decoder outputs the posterior distribution induced by the received messages, then expected log loss equals conditional entropy (Courtade et al., 2011).

This identity is the central reason the multiterminal problem becomes exactly characterizable. In the two-encoder model, the achievable rate–distortion region is the set of $x>0$4 for which there exist auxiliaries $x>0$5 satisfying

$x>0$6

$x>0$7

$x>0$8

and the paper proves this Berger–Tung-style region is exact (Courtade et al., 2011). The same phenomenon holds for the CEO problem under logarithmic loss, where the distortion becomes $x>0$9, and the paper extends the exact characterization to the $\log(1+u)$0-encoder CEO problem (Courtade et al., 2011).

This formulation makes “source representation” literal: the compressed description is evaluated by how well it supports a decoder-side posterior. The paper also links the framework to distributed approximation of the posterior $\log(1+u)$1, to list decoding, and to a relaxed Slepian–Wolf region in the special case $\log(1+u)$2, where the lossless bounds are shifted by the distortion level $\log(1+u)$3 (Courtade et al., 2011).

3. Log-domain data representations in neural computation

A different usage appears in low-precision neural computation. In “logarithmic data representation” for CNNs, a tensor value is stored through a quantized logarithm—typically base $\log(1+u)$4—so that reconstruction is a power of two and multiplication is replaced by bit-shifts or exponent addition (Miyashita et al., 2016). The paper is explicit that this is not a general-purpose source-coding theory, but a task-specific quantization and arithmetic scheme for CNN weights, activations, and gradients.

For inference, one variant encodes only one operand: $\log(1+u)$5 A second variant encodes both operands: $\log(1+u)$6 Zero is represented exactly, signs are carried separately when needed, and the paper defines an explicit elementwise quantizer $\log(1+u)$7 with clipping to a full-scale range (Miyashita et al., 2016).

The motivation is empirical and architectural. The paper argues that trained weights and ReLU activations are highly concentrated near zero, so uniform fixed-point quantization wastes representational density. In one reported activation-distribution experiment, the mean $\log(1+u)$8 quantization error was $\log(1+u)$9 for log quantization and $a_k=2^k(x^{2^{-k}}-1)$0 for linear quantization at the same bitwidth (Miyashita et al., 2016). The hardware consequence is that bulky digital multipliers can be eliminated.

The reported performance supports this interpretation. On ImageNet validation without retraining, VGG16 with log 3b activations achieved $a_k=2^k(x^{2^{-k}}-1)$1 top-5 accuracy, whereas linear 3b activations gave $a_k=2^k(x^{2^{-k}}-1)$2; log 4b activations reached $a_k=2^k(x^{2^{-k}}-1)$3, matching float32 (Miyashita et al., 2016). For convolutional weights, plain base-2 log quantization was fragile, but base-$a_k=2^k(x^{2^{-k}}-1)$4 log 5b substantially improved performance: on VGG16 it gave $a_k=2^k(x^{2^{-k}}-1)$5, compared with $a_k=2^k(x^{2^{-k}}-1)$6 for base-2 log 5b and $a_k=2^k(x^{2^{-k}}-1)$7 for linear 5b (Miyashita et al., 2016). In end-to-end low-precision training on CIFAR10, the log-quantized network achieved $a_k=2^k(x^{2^{-k}}-1)$8, compared with $a_k=2^k(x^{2^{-k}}-1)$9 for linear-5b and $\log x$0 for BinaryNet (Miyashita et al., 2016).

4. Operator-theoretic representations and logarithmic source conditions

In operator theory, logarithmic representation is used to encode generally unbounded infinitesimal generators through bounded logarithmic objects. One formulation assumes an evolution family $\log x$1 on a Banach space, a dense subspace $\log x$2, and commutativity between $\log x$3 and $\log x$4. The generator is then represented by

$\log x$5

with $\log x$6 as the bounded logarithmic representative (Iwata, 2017). The same paper shows that the logarithmic images can be organized into algebraic structures: $\log x$7 is a normed vector space, and $\log x$8 is a module over the Banach algebra $\log x$9 under the stated commutativity hypotheses (Iwata, 2017).

A related formulation introduces a weaker operator topology tailored to a single trajectory. In the locally-strong topology, $x-1-\log x$0 means $x-1-\log x$1 for one fixed $x-1-\log x$2, and the paper proves that this topology is weaker than the strong topology and not necessarily stronger than the weak topology (Iwata, 2020). Within that framework, the alternative infinitesimal generator

$x-1-\log x$3

is bounded, and the possibly unbounded generator is recovered by

$x-1-\log x$4

with $x-1-\log x$5 understood in the localized weak sense induced by locally-strong convergence (Iwata, 2020).

For unbounded evolution operators $x-1-\log x$6, the same program is extended by resolvent regularization. With

$x-1-\log x$7

the paper replaces $x-1-\log x$8 by logarithms of bounded resolvent transforms. In its one-log version,

$x-1-\log x$9

and the generator is represented as

$\sum 2^{-k}$0

under the stated resolvent assumptions (Iwata, 2021). The same paper also formulates a two-log representation using alternative infinitesimal generators $\sum 2^{-k}$1 and $\sum 2^{-k}$2 (Iwata, 2021).

In inverse problems, “logarithmic source representation” becomes a smoothness condition. For a bounded injective operator $\sum 2^{-k}$3 with nonclosed range, the paper studies solutions satisfying

$\sum 2^{-k}$4

equivalently $\sum 2^{-k}$5 with $\sum 2^{-k}$6 (Plato, 5 Sep 2025). This is explicitly described as weaker than a Hölder-type source condition $\sum 2^{-k}$7. For a broad class of regularization schemes,

$\sum 2^{-k}$8

and the resulting error rate has logarithmic order,

$\sum 2^{-k}$9

with both a priori and discrepancy-principle parameter choices analyzed (Plato, 5 Sep 2025). The paper also gives a concrete example for the integration operator $$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$0, showing that a borderline-smooth function of the form $$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$1, $$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$2, belongs to $$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$3 (Plato, 5 Sep 2025).

5. Nonlocal operators and variational source formulations

In PDE and nonlocal analysis, logarithmic source representation often means either an explicit singular-kernel formula for a logarithmic operator or a weak bilinear source formulation for the equation it defines. For the logarithmic Laplacian $$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$4, the Fourier symbol is

$$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$5

and the real-space formula is

$$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$6

with $$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$7 and $$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$8 (Chen et al., 2017). The paper treats the source problem

$$2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2 \sim \frac{(\log x)^2}{2}\,2^{-k},$$9

through the weak identity

$\log x<x-1$0

where $\log x<x-1$1 is the natural energy space. It emphasizes that no explicit Green kernel $\log x<x-1$2 is constructed; the available representation is variational and operator-theoretic rather than an explicit integral solution formula (Chen et al., 2017).

For the fractional logarithmic Schrödinger operator $\log x<x-1$3, the symbol is

$\log x<x-1$4

and the physical-space representation is the principal-value singular integral

$\log x<x-1$5

with

$\log x<x-1$6

built from the stable heat kernel (Feulefack, 2023). The paper proves

$\log x<x-1$7

for $\log x<x-1$8, and gives a whole-space Green representation

$\log x<x-1$9

for the Poisson problem $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$00 in $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$01, together with asymptotics of $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$02 at zero and infinity (Feulefack, 2023).

On weighted graphs, the logarithmic Laplacian is defined spectrally as $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$03 and admits the Bochner formula

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$04

in $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$05 (Chen et al., 8 Jul 2025). Under stochastic completeness, the pointwise representation becomes

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$06

with

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$07

The paper interprets this as a short-time symmetric nonlocal interaction term, a long-time source term, and an onsite correction. On weighted lattice graphs, it proves $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$08 and shows that $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$09 is not bounded on $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$10 (Chen et al., 8 Jul 2025).

For Schrödinger operators with potential, $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$11, the logarithmic operator is defined spectrally and also through the Frullani-type semigroup formula

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$12

in $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$13 (Betancor et al., 1 Apr 2026). When $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$14, $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$15, the pointwise representation is

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$16

where the nonconstant correction $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$17 reflects the non-Markovian fact that $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$18 (Betancor et al., 1 Apr 2026). The associated evolution problem

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$19

is solved by

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$20

for the stated data class (Betancor et al., 1 Apr 2026).

6. Defect, tensorial, and geometric extensions

A further extension arises from subdimensional disorder. The UV deformation

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$21

introduces a random coupling supported only on a $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$22-dimensional subspace, with Gaussian disorder

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$23

(Shimamori et al., 15 Oct 2025). The paper proposes that the disorder-averaged IR theory is a conformal defect in which bulk operators remain in ordinary conformal representations, while defect-local operators assemble into logarithmic multiplets. A rank-$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$24 defect multiplet is characterized by a Jordan block for the defect dilatation operator, and for rank $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$25 the finite transformation law is

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$26

(Shimamori et al., 15 Oct 2025).

The defect two-point functions are then logarithmic polynomials over power laws. For rank $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$27,

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$28

and bulk–defect correlators involve powers of

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$29

(Shimamori et al., 15 Oct 2025). In the free scalar with random pinning, the defect restriction $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$30 and the source field $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$31 form a rank-$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$32 logarithmic pair, and the replica analysis yields a half-line of fixed points

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$33

in codimension two (Shimamori et al., 15 Oct 2025). This is a direct instance in which the localized source itself becomes the logarithmic partner of a defect operator.

In continuum mechanics, logarithmic representation appears in a tensorial rather than defect-theoretic form. For the Hencky strain

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$34

the logarithmic spin tensor is represented by

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$35

where $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$36 is the commutator operator (Bathory et al., 22 Apr 2025). This replaces the classical eigenprojection formula by a commutator-based functional calculus. The paper emphasizes that if $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$37 and $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$38 commute, then $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$39 and $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$40 (Bathory et al., 22 Apr 2025).

A geometric variant appears in the representation of logarithmic connections by Fuchsian equations. A logarithmic connection on a rank-$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$41 bundle over $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$42 with parabolic points $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$43 can be represented by a scalar Fuchsian equation of order $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$44 whose prescribed singularities are the $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$45 and whose additional singularities are apparent (Ivanics, 2019). In rank $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$46, this gives a third-order equation and

$$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$47

apparent singularities, with coordinates $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$48 on a dense open subset of the moduli space (Ivanics, 2019). In this setting, “representation” means preservation of the monodromy representation up to conjugation, and logarithmic singularities of the connection are encoded as regular singular points of the scalar equation (Ivanics, 2019).

Taken together, these constructions show that “Logarithmic Source Representation” functions as a cross-disciplinary label for methods that move a problem into a logarithmic layer where structure becomes explicit: positivity for $$\log x=x-1-\sum_{k=1}^{\infty}2^{k-1}\bigl(x^{2^{-k}}-1\bigr)^2,$$49, posterior uncertainty under log loss, shift-based arithmetic in CNNs, bounded representatives of unbounded generators, singular-kernel source terms in nonlocal operators, or localized Jordan-block degrees of freedom on defects (Bradley, 2012, Courtade et al., 2011, Miyashita et al., 2016, Iwata, 2020, Chen et al., 2017, Shimamori et al., 15 Oct 2025).