In this paper, we introduce a novel method for predicting intraday instantaneous volatility based on Ito semimartingale models using high-frequency financial data. Several studies have highlighted stylized volatility time series features, such as interday auto-regressive dynamics and the intraday U-shaped pattern. To accommodate these volatility features, we propose an interday-by-intraday instantaneous volatility matrix process that can be decomposed into low-rank conditional expected instantaneous volatility and noise matrices. To predict the low-rank conditional expected instantaneous volatility matrix, we propose the Two-sIde Projected-PCA (TIP-PCA) procedure. We establish asymptotic properties of the proposed estimators and conduct a simulation study to assess the finite sample performance of the proposed prediction method. Finally, we apply the TIP-PCA method to an out-of-sample instantaneous volatility vector prediction study using high-frequency data from the S&P 500 index and 11 sector index funds.