Notation Reference — Time-Series JEPA

A standalone glossary for the Time-Series JEPA series, grouped by role, with a "read as" column. Keep it open in a second tab while reading.

The encoder is written \(E\) here, matching the Action Operators foundation and the Operator World Models series; it is the same object written \(\varphi_\psi\) in the I-JEPA literature.

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The signal and time

Symbol	Read as	Meaning
\(x\)	"x"	an observation — one timestep of the signal (in the running example, one day's behavioral index)
\(x_{\le t}\)	"x up to t"	the context window: the stretch of past signal ending at time \(t\)
\(x_{t+1}\)	"x at t plus 1"	the target: the next observation, the thing being predicted
\(t\)	"t"	the current time index
\(\Delta t\)	"delta t"	the prediction offset — how far ahead to predict (\(\Delta t = 1\) is one step)

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The four pieces

Symbol	Read as	Meaning
\(E_\xi\)	"E-xi"	the online encoder, trainable weights \(\xi\) (Greek xi); maps the past window to a latent
\(z_t = E_\xi(x_{\le t})\)	"z at t"	the context latent — a vector capturing where the system is now
\(E_{\bar\xi}\)	"E-xi-bar"	the target encoder: a slow EMA copy of \(E_\xi\), used to produce prediction targets; stop-gradient (no backprop into it)
\(\bar\xi \leftarrow \tau\bar\xi + (1-\tau)\xi\)	—	the EMA update of the target weights
\(\tau\)	tau	the EMA rate (e.g. \(0.99\)): how slowly the target encoder trails the online one
\(g_\phi\)	"g-phi"	the predictor, weights \(\phi\) (Greek phi); takes a context latent plus a query and returns the predicted target latent
\(q_{\Delta t}\)	"query at delta-t"	the query: tells the predictor what to predict; here it carries the offset \(\Delta t\) (how far ahead)
\(\hat z_{t+1}\)	"z-hat at t plus 1"	the predicted next latent, \(\hat z_{t+1} = g_\phi(z_t, q_{\Delta t})\)

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Loss and signals

Symbol	Read as	Meaning
\(\mathrm{sg}\)	"stop-grad"	stop-gradient: treat the argument as a fixed constant during backprop (applied to the target)
\(\mathcal{L}\)	"script L"	the training loss: latent-space squared error, \(\lVert \hat z_{t+1} - \mathrm{sg}(E_{\bar\xi}(x_{t+1})) \rVert^2\)
residual / surprise	—	the size of the prediction gap, \(\lVert z_{t+1} - \hat z_{t+1} \rVert\) — small when the signal follows its usual structure, large when it does not
\(\lVert v \rVert^2\)	—	squared Euclidean (\(\ell_2\)) norm
\(R(\omega)\)	"R of omega"	a rotation by angle \(\omega\) — the latent dynamics of a clean cycle in the running example (the gallery's rotation operator)

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Multimodal (Part 2)

Symbol	Read as	Meaning
\(m\)	"m"	a modality index — one channel of the signal (heart rate, sleep, phone activity, …)
\(E^{(m)}\)	"E-m"	a per-modality encoder for channel \(m\), before fusion into the shared latent \(z_t\)

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Series home: Time-Series JEPA. Start at Part 1 — From I-JEPA to Time-Series JEPA.