Multi-sample $$\zeta $$ -mixup: richer, more realistic synthetic samples from a p-series interpolant

Journal of Big Data

Table 1 Summary of notations

Notation	Description	Notation	Description
x	Input data sample	${\mathcal {H}}$	Entropy
y	Target label sample	${\mathcal {D}}$	Dimensionality of the input space
${\hat{x}}$	Synthesized input data sample	$\textrm{U}$	Uniform distribution
${\hat{y}}$	Synthesized target label sample	$\alpha $	mixup hyperparameter
${\mathcal {X}}$	Input data distribution	${\mathcal {M}}$	Data manifold
${\mathcal {Y}}$	Target label distribution	d	Intrinsic dimensionality of a manifold
P(x, y)	Data distribution over the input and the target	$N$	Number of samples in a dataset
$P_{\textrm{vic}} (x, y)$	Vicinal data distribution	$T$	Number of samples used for interpolation
${\mathcal {L}}$	Loss function	m	Number of samples in a mini-batch
R	Risk	$\pi $	${T} \times {T}$ random permutation matrix
$R_{\textrm{emp}}$	Empirical risk	s	Randomized ordering of samples
$R_{\textrm{vic}}$	Vicinal risk	$w_i$	Per-sample weight in $\zeta $-mixup
$\lambda $	Linear interpolation factor	C	Normalization constant for $\zeta $-mixup weights
${\mathcal {K}}$	Number of unique classes in the label distribution	$\gamma $	$\zeta $-mixup hyperparameter
${\mathcal {S}}$	Label space	$\gamma _{\textrm{min}}$	Minimum value of $\gamma $ to achieve the desirable properties of $\zeta $-mixup (see Theorem 1)

Notation	Description	Notation	Description
x	Input data sample	\({\mathcal {H}}\)	Entropy
y	Target label sample	\({\mathcal {D}}\)	Dimensionality of the input space
\({\hat{x}}\)	Synthesized input data sample	\(\textrm{U}\)	Uniform distribution
\({\hat{y}}\)	Synthesized target label sample	\(\alpha \)	mixup hyperparameter
\({\mathcal {X}}\)	Input data distribution	\({\mathcal {M}}\)	Data manifold
\({\mathcal {Y}}\)	Target label distribution	d	Intrinsic dimensionality of a manifold
P(x, y)	Data distribution over the input and the target	\(N\)	Number of samples in a dataset
\(P_{\textrm{vic}} (x, y)\)	Vicinal data distribution	\(T\)	Number of samples used for interpolation
\({\mathcal {L}}\)	Loss function	m	Number of samples in a mini-batch
R	Risk	\(\pi \)	\({T} \times {T}\) random permutation matrix
\(R_{\textrm{emp}}\)	Empirical risk	s	Randomized ordering of samples
\(R_{\textrm{vic}}\)	Vicinal risk	\(w_i\)	Per-sample weight in \(\zeta \)-mixup
\(\lambda \)	Linear interpolation factor	C	Normalization constant for \(\zeta \)-mixup weights
\({\mathcal {K}}\)	Number of unique classes in the label distribution	\(\gamma \)	\(\zeta \)-mixup hyperparameter
\({\mathcal {S}}\)	Label space	\(\gamma _{\textrm{min}}\)	Minimum value of \(\gamma \) to achieve the desirable properties of \(\zeta \)-mixup (see Theorem 1)