EASIER Notation: A Family of Simplified Formalisms

• EASIER Notation is a family of notational reforms aimed at reducing index bookkeeping and ambiguity in algebraic derivations.
• The Einstein-index variant streamlines tensor calculus in machine learning by using index-based representations to optimize differentiation and contraction ordering.
• The coordinate-free approach leverages explicit tensor products to clarify Fréchet derivative computations and higher-order derivative rules.

Searching arXiv for the cited papers to ground the article in the primary sources. arXiv search query: (Nakahira, 2022, Laue et al., 2020, Manton, 2012, Chiang et al., 2021, Rus, 15 Jan 2025) “EASIER Notation” is not a single universally standardized formalism. In the supplied arXiv literature, the label has at least two explicit meanings: an Einstein-index-based tensor-calculus representation for machine learning, and a combinatorial notation for Euler’s array of surjection counts. Several other papers use “easier” only descriptively, to characterize coordinate-free, named-axis, or graphical notations that reduce index bookkeeping, axis-order ambiguity, or symbolic overhead. A plausible implication is that the term is best read contextually: it denotes not one notation, but a family of notational reforms whose common goal is to make algebraic structure more visible and derivations less error-prone (Laue et al., 2020, Rus, 15 Jan 2025).

1. Terminological status and scope

The official status of the term varies sharply across sources. In "Yet another note on notation" (Rus, 15 Jan 2025), EASIER is an explicit acronym: Euler Array of Sub-power Integers with Exponential-like Recurrence. By contrast, "Differential Calculus, Tensor Products and the Importance of Notation" (Manton, 2012) states that the paper does not define “EASIER” as an acronym and uses “easier” only to emphasize pedagogical benefit. The recommender-systems notation paper likewise states that it does not introduce an acronym or named system called “EASIER,” and the VQEG SAM notation paper states that “EASIER” is not an official acronym there either (Ekstrand et al., 2019, Janowski et al., 2019).

This terminological dispersion matters because it prevents a naive identification of EASIER with a single symbolic language. In some contexts the label is attached to a concrete calculus with formal operators and algorithms; in others it is a retrospective editorial description of a notation that is simpler than dominant alternatives. The supplied sources therefore support a careful distinction between officially named notations and notations described as easier.

2. Einstein-index EASIER for tensor calculus in machine learning

In "A Simple and Efficient Tensor Calculus for Machine Learning" (Laue et al., 2020), EASIER notation is presented as Einstein-index-based Simple and Efficient Representation for tensor calculus in machine learning. Its central claim is that efficient higher-order differentiation does not require Ricci notation: Einstein notation suffices, while also aligning directly with the einsum-style contractions used in NumPy, TensorFlow, PyTorch, and JAX.

The formal basis is standard Einstein notation with a distinction between free indices and dummy indices. Free indices determine the shape of the result, while dummy indices are repeated and implicitly summed. The paper further formalizes a generic multiplication operator

$A *_{(s_1,s_2,s_3)} B,$

where $s_1$ and $s_2$ are input index sets and $s_3$ is the output index set. This covers outer products, matrix-vector products, matrix multiplication, and elementwise multiplication uniformly. The explicit output-index set is operationally important because it makes contractions first-class graph objects rather than ad hoc shorthand.

The computational model is an expression DAG whose nodes are binary multiplications, additions, and unary or elementwise functions. Derivatives are computed by forward-mode pushforwards, reverse-mode pullbacks, and a mixed cross-country mode: $$\frac{\partial y}{\partial x} = \sum_{v\in S} \bar{v} *_{(s_y s_v, s_v s_x, s_y s_x)} \dot{v},$$ where $S$ is a cut in the DAG. The stated purpose of cross-country mode is to reorder multiplications so that lower-order tensors are multiplied first, which is often preferable for non-scalar outputs.

A second key device is compression of higher-order derivatives. The paper states that suitable multiplication order can cancel or compact unit tensors, so that a fourth-order Hessian need not be materialized as such. Its matrix-factorization example compresses

$H \in \mathbb{R}^{n\times k\times n\times k}$

to a $k\times k$ matrix, changing solve time from $O((nk)^3)$ to $O(k^3)$. The empirical summary reports that the Einstein-notation method matches or exceeds the earlier Ricci-based approach, that logistic-regression Hessians are approximately $s_1$0 faster on CPU with cross-country mode, and that matrix-factorization and neural-net Hessians benefit substantially from compression. The implementation is stated to be available in www.MatrixCalculus.org.

Methodologically, this notation is “easier” not because it is informal, but because it stays in the same index language already used by ML frameworks while permitting a provably correct derivative algorithm for Jacobians and Hessians. That is a narrower and more technical claim than general readability: it is an argument about algorithmic efficiency, compatibility with existing AD systems, and contraction-order optimization.

3. Coordinate-free “easier” notation and tensor-aware $s_1$1-calculus

A different line of work appears in "Differential Calculus, Tensor Products and the Importance of Notation" (Manton, 2012). Here the relevant notation is coordinate-free rather than index-based. Its central object is the Fréchet derivative $s_1$2, treated as a linear map from directions in the domain to changes in the codomain. Higher derivatives are multilinear maps, and the paper’s “small but crucial modification” is to use tensor products explicitly so that direction arguments are routed correctly.

The basic definitions are: $s_1$3 with the canonical identification

$s_1$4

This produces a typed calculus in which ambiguity in repeated chain and product rules is removed by explicit tensor structure. The tensor-aware chain rule is written

$s_1$5

and the corrected operator-valued product rule is

$s_1$6

The paper also gives higher-order chain rules such as [ D^2(f\circ g) = (D^2 f \circ g)(Dg\otimes Dg) + (Df\circ g)D^2